text:
\emph{Throughout the article, the base field is an algebraically closed field $k$ of characteristic zero.}

\subsection{Boundary free statements}
\label{sec:intro_boundary_free}

For a flat family of $n$-dimensional Fano family $f : X \to T$ (that is, $X$ and $T$ are normal and projective, and $-K_{X/T}$ is an $f$-ample $\bQ$-Cartier divisor) the Chow-Mumford (CM) line bundle is the pushforward cycle
\begin{equation}
\label{eq:CM_definition_intro}
\lambda_f := - f_* \left( c_1(-K_{X/T})^{n+1}\right).
\end{equation}
This cycle, up to multiplying with a positive rational number, is the first Chern class of the
a functorial line bundle on $T$ defined in  \cite{Paul_tian1,Paul_tian2} (see also \cite{Schumacher,Fine_Ross_A_note_on_positivity_of_the_CM_line_bundle,PRS}).

Our main motivation to consider  the CM line bundle originates from the classification theory of algebraic varieties: the \emph{birational part} of classification theory, also called the Minimal Model Program, predicts that up to specific birational equivalences, each projective variety decomposes into iterated fibrations with general fibers of 3 basic types: Fano, weak Calabi-Yau, and general type (to be precise one here needs to allow pairs, see \autoref{sec:intro_logarithmic}, but the boundary free case is a good first approximation).  These 3 types are defined by having a specific class of mild singularities and negative/numerically trivial/positive canonical bundles. Then the \emph{moduli part} of the classification theory is supposed to construct compactified moduli spaces for the above 3 basic types of varieties. According to our current understanding the moduli part seems to be doable only in the presence of a (singular) K\"ahler-Einstein metric,  which is predicted to be equivalent to the algebraic notion of $K$-polystability.

We note that the above predictions are proven in large cases, e.g., MMP: \cite{Birkar_Cascini_Hacon_McKernan_Existence_of_minimal_models,Hacon_McKernan_Existence_of_minimal_models_for_varieties_of_log_general_type_II,
Hacon_Xu_Existence_of_log_canonical_closures,
Birkar_Existence_of_log_canonical_flips_and_a_special_LMMP,
Fujino_Introduction_to_the_log_minimal_model_program_for_log_canonical_pairs,
flips_and_abundance_for_algebraic_threefolds,Keel_Matsuki_McKernan_Log_abundance_theorem_for_threefolds,
Alexeev_Hacon_Kawamata_Termination_of_many_4-dimensional_log_flips,
Birkar_On_termination_of_log_flips_in_dimension_four,Birkar_Ascending_chain_condition_for_log_canonical_thresholds_and_termination_of_log_flips}; connections between $K$-stability and K\"ahler-Einstein metrics: \cite{Berman_Guenancia_Kahler-Einstein_metrics_on_stable_varieties_and_log_canonical_pairs,Chen_Donaldson_Sun_Kahler-Einstein_metrics_on_Fano_manifolds_I_Approximation_of_metrics_with_cone_singularities,
Chen_Donaldson_Sun_Kahler-Einstein_metrics_on_Fano_manifolds_II_Limits_with_cone_angle_less_than_2pi,
Chen_Donaldson_Sun_Kahler-Einstein_metrics_on_Fano_manifolds_III_Limits_as_cone_angle_approaches_2pi_and_completion_of_the_main_proof,
Tian_K-stability_and_Kahler-Einstein_metrics,Odaka_The_GIT_stability_of_polarized_varieties_via_discrepancy,Odaka_The_Calabi_conjecture_and_K-stability,Odaka_Xu_Log-canonical_models_of_singular_pairs_and_its_applications, Recent_Tian}.

In particular, on the Fano side for the moduli part one should construct algebraically a moduli space conjectured below. Furthermore, the polarization on this moduli space should be given by the CM line bundle, as the connection between the K\"ahler-Einstein theory and $K$-stability originates from Tian's idea to view the metrics as polystable objects in an infinite dimensional GIT type setting with respect to the CM line bundle.  (See \autoref{def:K_stability} and \autoref{def:K_stable} for a precise definition and characterization used in the present article for $K$-semistability and see \autoref{sec:K_stability_versions} for an explanation on $K$-polystability). The precise conjecture is as follows:

\begin{conjecture}{\rm (e.g., a combination of  \cite[Conj 3.1]{Odaka_On_the_moduli_of_Kahler-Einstein_Fano_manifolds} and \cite[Qtn 3, page 15]{Odaka_Compact_moduli_space_of_Kahler_Einstein_Fano_varieties}, but also implicit to different degrees for example in \cite{Tian2,Paul_tian2,Chen_Donaldson_Sun_Kahler-Einstein_metrics_on_Fano_manifolds_III_Limits_as_cone_angle_approaches_2pi_and_completion_of_the_main_proof,Li_Wang_Xu_Qasi_projectivity_of_the_moduli_space_of_smooth_Kahler_Einstein_Fano_manifolds,Li_Wang_Xu_Degeneration,SpottiYTD})}
\label{conj:K_ss_moduli}
\begin{enumerate}
\item \label{itm:K_ss_moduli:stack}  The stack $\sM_{n,v}^{\Kss}$ of $K$-semistable  klt Fano varieties of fixed dimension $n>0$ and anti-canonical volume $v>0$  is an Artin stack of finite type over $k$.

\item \label{itm:K_ss_moduli:algebraic_space} The stack $\sM_{n,v}^{\Kss}$ admits a good moduli space $M^{\Kps}_{n,v}$ (in the sense of \cite{Alper_Good_moduli_spaces_for_Artin_stacks}), which is proper over $k$, and which parametrizes $K$-polystable klt Fano varieties of dimension $n$ and volume $v$. Furthermore,  the CM line bundle descends onto $M^{\Kps}_{n,v}$.
 \item \label{itm:K_ss_moduli:polarization} $M^{\Kps}_{n,v}$  is a projective scheme via the polarization given by the CM line bundle. In particular, the CM line bundle is
 \begin{enumerate}
 \item  \label{itm:K_ss_moduli:semi_positive} nef in families of $K$-semi-stable klt Fanos, and
 \item \label{itm:K_ss_moduli:positive} nef and big in maximally varying families of $K$-polystable klt Fanos.
 \end{enumerate}

\end{enumerate}
\end{conjecture}

Our main result concerns point \autoref{itm:K_ss_moduli:polarization} of \autoref{conj:K_ss_moduli}. We completely solve point \autoref{itm:K_ss_moduli:semi_positive}  and we solve  \autoref{itm:K_ss_moduli:positive} on the uniformly $K$-stable locus (Notes: $K$-polystability usually occurs at the boundary of the uniformly $K$-stable locus, see \autoref{sec:K_stability_versions} for the definitions; see also \autoref{rem:why_only_unif_K_stable} for the reasons of the specific generality in point \autoref{itm:K_ss_moduli:positive}, and see \autoref{sec:intro_logarithmic} for the results for pairs).

\begin{theorem}
\label{thm:semi_positive_no_boundary}
Let $f : X \to T$ be a flat morphism with connected fibers between normal projective varieties such that $-K_{X/T}$ is $\bQ$-Cartier and $f$-ample, and let $\lambda_f$ be the CM line bundle defined in equation \autoref{eq:CM_definition_intro}.
\begin{enumerate}
\item \label{itm:semi_positive_no_boundary:pseff} {\scshape Pseudo-effectivity:} If $T$ is smooth and the very general geometric fibers of $f$ are $K$-semistable, then {$\lambda_f$ is pseudo-effective}.
\item \label{itm:semi_positive_no_boundary:nef} {\scshape Nefness:} If all the geometric fibers of $f$ are $K$-semi-stable then {$\lambda_f$ is nef}.
\item \label{itm:semi_positive_no_boundary:big} {\scshape Bigness:} If $T$ is smooth, the very general geometric fibers of $f$ are uniformly $K$-stable,  the variation of $f$ is maximal (i.e., there is a non-empty open set of $T$ over which the isomorphism equivalence classes of the fibers are finite), and either $\dim T=1$ or the fibers of $f$ are reduced, then {$\lambda_f$ is big}.
\item \label{itm:semi_positive_no_boundary:ample} {\scshape Ampleness:} If all the geometric fibers of $f$ are uniformly $K$-stable and the isomorphism equivalence classes of the fibers are finite, then {$\lambda_f$ is ample}.
\item \label{itm:semi_positive_no_boundary:q_proj} {\scshape Quasi-projectivity:} If $T$ is only assumed to be a proper normal algebraic space, all the geometric fibers are $K$-semi-stable and there is an open set $U \subseteq T$ over which the geometric fibers are uniformly $K$-stable and the isomorphism classes of the fibers are finite, then $U$ is a quasi-projective variety.
\end{enumerate}
\end{theorem}

\begin{remark}
Notably, \autoref{thm:semi_positive_no_boundary} deals with non-smoothable singular Fanos too, about which we remark that:
\begin{enumerate}
\item This is the first result about (semi-)positivity of the CM line bundle dealing with non-smoothable singular Fanos, as explained in \autoref{rem:partial_results}.
\item Non-smoothable singular Fanos is the general case, for example in the sense that smoothable $K$-semi-stable Fanos are bounded, even without fixing the anti-canonical volume (one can put this together from \cite{Kollar_Miyaoka_Mori_Rational_connectedness_and_boundedness_of_Fano_manifolds,Jiang_Boundedness_of_Q-Fano_varieties_with_degrees_and___alpha-invariants_bounded_from_below}). On the other hand, non-smoothable $K$-semi-stable Fanos are unbounded if one does not fix the volume, as can be seen by considering quasi-\'etale quotients by bigger and bigger finite groups of $\bP^2$, which are $K$-semi-stable according to \cite[Cor. 1.7]{Fujita_Uniform_K-stability_and_plt_blowups_of_log_Fano_pairs}.

\end{enumerate}

\end{remark}

\begin{remark}
The proof of \autoref{thm:semi_positive_no_boundary} uses the Central Limit Theorem of probability theory. See \autoref{sec:outline_semi_positivity} for an outline of our argument or \autoref{prop:vectP2} for the precise place where the Central Limit Theorem is used.
\end{remark}

\begin{remark}
Either all or no very general geometric fibers is $K$-semi-stable (resp. uniformly $K$-stable), as shown by the constancy of the $\delta$-invariant on very general generic fibers (\autoref{prop:delta_general_fiber}). In particular,  if $k$ is uncountable, say $\bC$, the assumptions of \autoref{thm:semi_positive_no_boundary} can be checked on closed fibers. Furthermore, in the $K$-semistable cases, so for points \autoref{itm:semi_positive_no_boundary:pseff} and \autoref{itm:semi_positive_no_boundary:nef} of \autoref{thm:semi_positive_no_boundary}, it is enough to find a single $K$-semistable closed fiber, according to \cite[Thm 3]{Blum_Liu_The_normalized_volume_of_a_singularity_is_lower_semicontinuous}. The uniformly $K$-stable version of \cite[Thm 3]{Blum_Liu_The_normalized_volume_of_a_singularity_is_lower_semicontinuous} is not known, but it is expected too.

We also remark that in \autoref{thm:semi_positive_no_boundary} we carefully said ``geometric fiber'' instead of just ``fiber''. The reason is that we use the $\delta$-invariant description of $K$-stability, and the $\delta$-invariant of a variety is not invariant under base extension to the algebraic closure (see \autoref{rem:delta_invariant_non_albraically_closed_field}). So, for scheme theoretic fibers over non algebraically closed  fields the $\delta$-invariant can have non semi-continuous behavior.
\end{remark}

\begin{remark}
We chose the actual generality for \autoref{thm:semi_positive_no_boundary}, as it is the generality in which the relative canonical divisor exists and admits reasonable base-change properties (see \autoref{sec:relative_canonical} for details) on very general curves in moving families of curves on the base.

Nevertheless, in situations where this base-change is automatic, \autoref{thm:semi_positive_no_boundary} directly implies statements over non-normal, non-projective, and even non-scheme bases. This is made precise in the following statement:
\end{remark}

\begin{corollary}
\label{cor:proper_base}
Let $f : X \to T$ be a flat, projective morphism with connected fibers to a proper algebraic space, such that there is an integer $m > 0$ for which  $\omega_{X/T}^{[m]}$  is a line bundle and all  the geometric fibers are $K$-semi-stable klt Fano varieties. Let $N$ be the CM-line bundle associated to the polarization $\omega_{X/T}^{[-m]}$ as defined over general bases in \cite{Paul_tian2} (see \autoref{notation:Paul_Tian}).
Then, $N$ is nef, and if the variation of $f$ is maximal and the very general geometric fiber is uniformly $K$-stable, then $N$ is big.

\end{corollary}

\begin{remark}
Note that over $\bC$ the positivity properties of \autoref{thm:semi_positive_no_boundary} (nefness, pseudo-effectivity, bigness, ampleness) can be characterized equivalently analytically, e.g., \cite[Prop 4.2]{Demailly_Singular_Hermitian_metrics_on_positive_line_bundles}
\end{remark}

\begin{remark}
{\scshape Negativity of $-K_{X/T}$ point of view.}
Unwinding definition \autoref{eq:CM_definition_intro},
we obtain that \autoref{thm:semi_positive_no_boundary} in the case of one dimensional base states that $(-K_{X/T})^{n+1}$ is at most zero/smaller than 0. Using this in conjunction with the base-change property with the CM line bundle (\autoref{prop:CM_base_change}) we obtain that \autoref{thm:semi_positive_no_boundary}, especially the last 3 points, prove strong negativity property of $-K_{X/T}$ for klt Fano families.

There does exist birational geometry statements claiming that $-K_{X/T}$ is not nef, e.g., \cite[Prop 1]{Zhang_On_projective_manifolds_with_nef_anticanonical_bundles}. Our negativity statements point in this direction but go further. However, it is not a coincidence that strong negativity statements on $-K_{X/T}$ did not show up earlier, as in fact \autoref{thm:semi_positive_no_boundary} is not true for every family of klt Fano varieties. Indeed, \autoref{ex:negative_degree} shows that   in  \autoref{thm:semi_positive_no_boundary} one cannot relax the $K$-semi-stable Fano assumption to just assuming klt Fano.
The development of the notions of $K$-stability in the past decade was essential for creating the chance of proving negativity statements for $-K_{X/T}$ of the above type.

We also note that as $-K_{X/T}$ is not nef usually in the above situation (c.f., \autoref{thm:nef_threshold} and \autoref{ex:not_nef}), the negativity of $(-K_{X/T})^{n+1}$ is independent of the negativity of $\kappa(-K_{X/T})$. In fact,  assuming the former, $\kappa(-K_{X/T})$ can be $- \infty$ (\autoref{ex:anti_canonical_no_section}), 0 (\autoref{ex:not_nef}), $\dim X$ (\autoref{ex:positive_and_big}), and also something in between the latter two values (\autoref{ex:anti_canonical_no_section}).
\end{remark}

\begin{remark}
\label{rem:partial_results}
The following are the already known partial results on  \autoref{conj:K_ss_moduli}.
\begin{enumerate}
\item On the algebraic side, aiming for all klt singularities, there were no results on point \autoref{itm:K_ss_moduli:polarization} of \autoref{conj:K_ss_moduli} earlier (although the second author with Xu proved the canonically polarized version in \cite{Patakfalvi_Xu_Ampleness_of_the_CM_line_bundle_on_the_moduli_space_of_canonically_polarized_varieties}). Speaking about points \autoref{itm:K_ss_moduli:stack} and  \autoref{itm:K_ss_moduli:algebraic_space} of \autoref{conj:K_ss_moduli}, they decompose into statements about different properties of the moduli functor: boundedness, separatedness, properness, openness of K-semi-stability,  and contrary to the canonically polarized case, an analysis of the action of automorphisms on  the CM-line bundle is also necessary to have it descend to the coarse moduli space. Out of these, only boundedness is known according to  \cite[Cor 1.7]{Jiang_Boundedness_of_Q-Fano_varieties_with_degrees_and___alpha-invariants_bounded_from_below}, which uses also crucially the seminal papers of Birkar \cite{Birkar_Anti-pluricanonical_systems_on_Fano_varieties,Birkar_Singularities_of_linear_systems_and_boundedness_of_Fano_varieties}.

 \item On the other hand, using analytic methods (sometimes in conjunction with algebraic ones), there are plenty of results about \autoref{conj:K_ss_moduli} on the closure of the locus of smooth Fanos:
 \cite{Li_Wang_Xu_Qasi_projectivity_of_the_moduli_space_of_smooth_Kahler_Einstein_Fano_manifolds,Li_Wang_Xu_Degeneration,SpottiYTD,Odaka_Compact_moduli_space_of_Kahler_Einstein_Fano_varieties}.

 The only significant piece missing from the above results is that the positivity of the CM line bundle is not known on closed subspaces $V$ lying in the boundary of the closure of the locus of smooth Fanos. Our theorem in particular remedies this if the very general Fano parametrized by $V$ is uniformly $K$-stable (and necessarily singular), see \autoref{cor:bigger_open_set}.
\end{enumerate}

\end{remark}

\begin{remark}
\label{rem:why_only_unif_K_stable}
There are two main reasons why our positivity statements (points \autoref{itm:semi_positive_no_boundary:big}, \autoref{itm:semi_positive_no_boundary:ample} and \autoref{itm:semi_positive_no_boundary:q_proj} of \autoref{thm:semi_positive_no_boundary}) work in the uniformly $K$-stable case, but not in the $K$-polystable case:
\begin{enumerate}
\item We rely on the characterization of $K$-semistability and uniform $K$-stability via the $\delta$ invariant given by \cite{Fujita_Odaka_On_the_K-stability_of_Fano_varieties_and_anticanonical_divisors,BJ}. Such characterization is not available for the $K$-polystable case.
\item Our theorem on the nef threshold (\autoref{thm:nef_threshold} below, on which the above 3 points of \autoref{thm:semi_positive_no_boundary} depend) fails in the $K$-polystable case according to \autoref{ex:not_nef}. Hence, one would need a significantly different approach to extend points \autoref{itm:semi_positive_no_boundary:big}, \autoref{itm:semi_positive_no_boundary:ample} and \autoref{itm:semi_positive_no_boundary:q_proj} of \autoref{thm:semi_positive_no_boundary} to the $K$-polystable case.
\end{enumerate}

\end{remark}

\begin{remark}
One could make definition \autoref{eq:CM_definition_intro} also without requiring flatness. We do not know if \autoref{thm:semi_positive_no_boundary} holds in this situation. Nevertheless, we note that it would be interesting to pursue this direction for example for applications to Mori-fiber spaces with higher dimensional bases (see \autoref{cor:bounding_volume}).

Also we expect that the reduced fiber assumption of point \autoref{itm:semi_positive_no_boundary:big} of \autoref{thm:semi_positive_no_boundary} can be removed, as we needed it for technical reasons (certain base changes over movable curves are nice), and also because the conjectured  $K$-semi-stable reduction should eliminate it.
\end{remark}

\subsection{Logarithmic statements}
\label{sec:intro_logarithmic}

We prove points \autoref{itm:semi_positive_no_boundary:pseff} and \autoref{itm:semi_positive_no_boundary:nef} of \autoref{thm:semi_positive_no_boundary} in the generality of pairs. We  state this separately, in the present subsection, as the statements are more cumbersome  (e.g., one needs to guarantee that the boundary can restrict to fibers, etc.).

If $f : (X, \Delta) \to T$ is a flat morphism of relative dimension $n$ from a projective normal pair to a normal projective variety such that $-(K_{X/T} + \Delta)$ is $\bQ$-Cartier and $f$-ample. Then we define the CM line bundle by
\begin{equation}
\label{eq:CM_line_bundle_log}
\lambda_{f, \Delta}:= -f_* (-(K_{X/T} + \Delta)^{n+1} ).
\end{equation}

\begin{theorem}
\label{thm:semi_positive_boundary}
Let $f : X \to T$ be a flat morphism of relative dimension $n$ with connected fibers between normal projective varieties and let $\Delta$ be an effective $\bQ$-divisor on $X$  such that $-(K_{X/T} + \Delta)$ is $\bQ$-Cartier and $f$-ample. Let $\lambda_{f,\Delta}$ be the CM  line bundle on $T$ as defined in \autoref{eq:CM_line_bundle_log}.
\begin{enumerate}
\item {\scshape Pseudo-effectivity:} \label{itm:semi_positive_boundary:pseff} If $T$ is smooth and  $(X_t, \Delta_t)$ is K-semi-stable  for  very general geometric fibers $X_t$, then $\lambda_{f, \Delta}$ is pseudo-effective.
\item {\scshape Nefness:} \label{itm:semi_positive_boundary:nef} If all fibers $X_t$ are normal, $\Delta$ does not contain any fibers (so that we may restrict $\Delta$ on the fibers), and $(X_t, \Delta_t)$ is $K$-semi-stable for all geometric fibers $X_t$, then $\lambda_{f, \Delta}$ is nef.
\end{enumerate}
\end{theorem}

\begin{remark}
\label{rem:no_log_pos}
We note that positivity statements, that is, points \autoref{itm:semi_positive_no_boundary:ample}, \autoref{itm:semi_positive_no_boundary:big} and \autoref{itm:semi_positive_no_boundary:q_proj} of \autoref{thm:semi_positive_no_boundary}, are much trickier to prove in the logarithmic case than in the boundary free case (c.f. \cite{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension}). The main issue is that a family of pairs can have maximal variations in a way that the underlying family of varieties has no variation, but  at the same time the underlying family of varieties is not trivializable on a finite cover (e.g., take the family of \autoref{ex:negative_degree} and put a pair structure on it using a small multiple of varying anti-pluricanonical divisors). The same issue in the canonically polarized case involved a decent amount of work \cite{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension,Patakfalvi_Xu_Ampleness_of_the_CM_line_bundle_on_the_moduli_space_of_canonically_polarized_varieties}.

\end{remark}

\subsection{Applications}
\label{sec:intro_applications}

Our applications of the theorems above are of two type:
\begin{enumerate}
 \item moduli theoretic, as we have already suggested in \autoref{sec:intro_boundary_free}, and
 \item birational geometric, aiming to understand  Mori-fiber spaces with $K$-semistable fibers.
\end{enumerate}
We start with the precise statements of the moduli theoretic ones (\autoref{cor:moduli} and \autoref{cor:bigger_open_set}):

\begin{corollary}
\label{cor:moduli}
If $M$ is a proper algebraic space which is the  moduli space for some class $\fX$ of $K$-semi-stable Fanos with the uniform $K$-stable locus $M^u \subseteq M$ being open (see \autoref{def:moduli} for the precise definitions), then the normalization of $M^u$ is a quasi-projective scheme over $k$.
\end{corollary}

\begin{remark}
The space $M^u$ of \autoref{cor:moduli} is many times smooth already, in which case the normalization can be certainly dropped from the statement. In fact, we know that it is smooth at the points corresponding to smooth Fanos
\cite{Ran_Deformations_of_manifolds_with_torsion_or_negative_canonical_bundle,Kawamata_Unobstructed_deformations_A_remark_on_a_paper_of_Z_Ran}, and to terminal Fano $3$-folds
\cite[Thm 1.7]{Sano_On_deformations_of_Q-Fano_3-folds}. Unfortunately, these unobstructedness statements do not hold for all Fanos, as \cite[Rem 2.13]{Sano_On_deformations_of_Q-Fano_3-folds} gives a counterexample. However, the counterexample is a cone over a Del-Pezzo surface of degree 6. Hence, it has infinite automorphism group, and in particular it is not uniformly $K$-stable. This leads to the following question.
\end{remark}

\begin{question}
 Is the deformation space of uniformly $K$-stable Fanos (including general klt ones) unobstructed?
\end{question}

In the next corollary, we extend the locus of the moduli space of $K$-semistable Fanos that is known to be quasi-projective from the smooth locus to the union of the smooth locus and the largest open set in the  uniformly $K$-stable locus.

\begin{corollary}
\label{cor:bigger_open_set}
If $M$ is the moduli space of smoothable $K$-semistable  Fanos (which is known to exist as an algebraic space according to \cite{Li_Wang_Xu_Qasi_projectivity_of_the_moduli_space_of_smooth_Kahler_Einstein_Fano_manifolds,Li_Wang_Xu_Degeneration,SpottiYTD,Odaka_Compact_moduli_space_of_Kahler_Einstein_Fano_varieties}), and $M^0$ is an open set parametrizing Fanos that are either smooth or uniformly $K$-stable, then the normalization of $M^0$ is quasi-projective.
\end{corollary}

Fujita showed in \cite[Thm 1.1]{Fujita_Optimal_bounds_for_the_volumes_of_Kahler-Einstein_Fano_manifolds} that $\vol(-K_X) \leq (n+1)^n$ for every $K$-semistable Fano variety $X$ of dimension $n$ (see \cite[Thm 3]{Liu_The_volume_of_singular_Kahler-Einstein_Fano_varieties} for better bounds in the presence of quotient singularities). Using \autoref{thm:semi_positive_boundary} we can show similar bounds for (non-necessarily klt) Fano $X$ admitting a  Fano fibration structure with $K$-semi-stable general fiber.

\begin{corollary}
\label{cor:bounding_volume}
If $(X,\Delta)$ is a normal Fano pair, and $f : X \to \bP^1$ is a fibration with $K$-semi-stable very general geometric fibers $F$, then
\begin{equation*}
\vol(-(K_X+\Delta)) \leq 2 \dim  \left( X\right) \vol (-(K_F+\Delta_F)).
\end{equation*}
 If furthermore $\Delta=0$, then
 \begin{equation*}
 \vol(-K_X) \leq 2\dim \left(X\right)^{\dim \left(X\right)}.
 \end{equation*}
\end{corollary}

\begin{remark}
\autoref{cor:bounding_volume} is sharp  for surfaces and threefolds. Indeed, a del Pezzo surface of degree $8$ and the blow-up of $\bP^3$ at a line (whose anti-canonical volume is $54$) can be fibred over $\bP^1$ with $K$-semis-table fibres.
\end{remark}

\begin{remark}
\label{rem:K_stability_classification}
{ \scshape Classification of (uniform) $K$-(semi/poly)-stable Fanos}: to understand the power of \autoref{cor:bounding_volume}, it is useful to know which Fanos are $K$-semi-stable and which are not. In fact, then one want to figure this out for all the four $K$-stability properties (see \autoref{sec:K_stability_versions}), which has been an active area of research recently. To start with, let us recall that $K$-semi-stable Fano varieties are always klt.

A Del-Pezzo surface is K-polystable if and only if it is not  of degree $8$ or $7$ \cite{Tian_Yau_Kahler-Einstein_metrics_on_complex_surfaces_with_C_1_at_least_zero,Tian_On_Calabis_conjecture_for_complex_surfaces_with_positive_first_Chern_class}.
Smooth Fano surfaces with discrete automorphism groups are even uniformly K-stable, and their delta invariant (see \autoref{sec:delta}) is bounded away from $1$ in an effective way \cite{PW}. Smoothable singular K-stable Del-Pezzo surfaces are classified in \cite{Odaka_Spotti_Sun}.

K-stable proper intersection of two quadrics in an odd dimensional projective space are classified in \cite{Spotti_Sun} (also \cite{Arezzo_Ghigi_Pirola_Symmetries_quotients_and_Kahler-Einstein_metrics}); in particular, smooth varieties of these types are always K-stable. Cubic $3$-folds are studied in \cite{Li_Xu}, where again smooth ones are $K$-stable, and so are the ones containing only $A_k$ singularities for $k \leq 4$. Under adequate hypotheses, in \cite{Dervan_Finite}, it is shown that Galois covers of K-semistable Fano varieties are K-stable. This can be applied for instance to double solids. Furthermore, birational superigid Fano varieties are K-stable under some addition mild hypothesis \cite{OO13,SZ,Z}. However, according to the best knowledge of the authors, there is not a complete classification of K-stable smooth Fano threefolds.

If one wants to study klt Fano varieties from the point of view of the MMP, it is particularly relevant to see if one can apply \autoref{cor:bounding_volume} to the case of Mori Fibre Spaces. In \cite[Corollary 1.11]{CFST}, it is shown that if a smooth Fano surface or a smooth toric variety can appear as a fibre of MFS, then it is K-semistable. We do not know if the analogous  result holds in dimension $3$. However, there are examples of smooth Fano fourfolds with Picard number one (which then can be general fibers of  MFS's) that are not K-semistable \cite{Fujita_Examples}, see also \cite{Codogni2018}.

\end{remark}

\subsection{Byproduct statements}

As a byproduct of our method we obtain the following bound on the nef threshold of $-(K_{X/T} + \Delta)$ with respect to $\lambda$ in the uniformly $K$-stable case.

\begin{theorem}
\label{thm:bounding_nef_threshold}
\label{thm:nef_threshold}
Let $f : X \to T$ be a flat morphism with connected fibers from a normal projective variety of dimension $n+1$ to a smooth curve and let $\Delta$ be an effective $\bQ$-divisor on $X$  such that
\begin{itemize}
 \item $-(K_{X/T} + \Delta)$ is $\bQ$-Cartier and $f$-ample, and
 \item $\left(X_{\ot},\Delta_{\ot}\right)$ is uniformly $K$-stable for very general geometric fibers $X_{\ot}$.
\end{itemize}

Set
\begin{itemize}

 \item \label{itm:thm:bounding_nef_threshold_Delta} set $\delta:=\delta\left(X_{\ot}, \Delta_{\ot}\right)$ for $\ot$ very general geometric point, and

 \item let $v:=\left( (-K_{X/T} - \Delta)_t \right)^n$ for any $t \in T$.
\end{itemize}
Then, $- K_{X/T} - \Delta + \frac{\delta}{(\delta -1) v (n+1)} f^* \lambda_{f,\Delta} $ is nef.
\end{theorem}

\begin{remark}
One cannot have a nef threshold statement as   in \autoref{thm:nef_threshold} for uniformly $K$-stable replaced with $K$-polystable. Indeed, take the family $f : X \to T$ given by  \autoref{ex:not_nef}. It has $K$-polystable fibers, $\deg \lambda_f=0$, but $K_{X/T}$ is not nef. In particular, for any $a \in \bQ$,  $-K_{X/T}+ a f^*\lambda_f \equiv K_{X/T}$, and hence for any $a \in \bQ$, $-K_{X/T}+ a f^*\lambda_f$ is not nef.

\end{remark}

And we have a structure theorem when $\lambda$ is not positive:

\begin{theorem}
\label{thm:lambda_non_big}
Let $f : X \to T$ be a flat morphism of relative dimension $n$ with connected fibers between normal projective varieties and let $\Delta$ be an effective $\bQ$-divisor on $X$  such that $-(K_{X/T} + \Delta)$ is $\bQ$-Cartier and $f$-ample. Assume that $\left(X_{\ot}, \Delta_{\ot}\right)$ is uniformly $K$-stable for very general geometric fibers $X_{\ot}$. If $H$ is an ample divisor on $T$, such that  $\lambda_{f,\Delta} \cdot H^{\dim T -1} =0$, then  for every integer $q>0$ divisible enough,  $f_* \sO_X(q (-K_{X/T} - \Delta))$ is an $H$-semi-stable vector bundle of slope $0$.
\end{theorem}

\subsection{Similar results in other contexts}

Roughly, there are three types of statements above: (semi-)positivity results, moduli applications, inequality of volumes of fibrations. Although in the realm of $K$-stability ours are the first general algebraic results, statements of these types were abundant in other, somewhat related, contexts: KSBA stability, GIT stability, and just general algebraic geometry. Our setup and our methods are different from these results, still we briefly list some of them for completeness of background. We note that KSBA stability is related to our framework as it is shown to be exactly the canonically polarized $K$-stable situation \cite{Odaka_The_GIT_stability_of_polarized_varieties_via_discrepancy,Odaka_The_Calabi_conjecture_and_K-stability,Odaka_Xu_Log-canonical_models_of_singular_pairs_and_its_applications}. Also, GIT stability is related, as $K$-stability originates from an infinite dimensional GIT, although it is shown that it cannot be reproduced using GIT (e.g., \cite{Wang_Xu_Nonexistence_of_asymptotic_GIT_compactification}).

\noindent
\begin{tabular}{|l|p{110pt}|p{110pt}|p{77pt}|}
\hline
& general algebraic geometry & KSBA stability & GIT stability \\
\hline
(semi-)positivity & \cite{Griffiths_Periods_of_integrals,Fujita_On_Kahler_fiber_spaces,Kawamata_Characterization_of_abelian_varieties,Viehweg_Weak_positivity,Kollar_Subadditivity_of_the_Kodaira_dimension} & \cite{Kollar_Projectivity_of_complete_moduli,Fujino_Semi_positivity_theorems_for_moduli_problems,Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension,Patakfalvi_Xu_Ampleness_of_the_CM_line_bundle_on_the_moduli_space_of_canonically_polarized_varieties} & \cite{CH} \\
\hline
moduli applications & \cite{Viehweg_Quasi_projective_moduli} & \cite{Kollar_Projectivity_of_complete_moduli,Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension,Ascher_Bejleri_Moduli_of_weighted_stable_elliptic_surfaces_and_invariance_of_log___plurigenera} & \cite{CH} \\
\hline
volume (slope) inequalities & \cite{Xiao_Fibered_algebraic_surfaces_with_low_slope} &  & \cite{Pardini,Stoppino}\\
\hline
\end{tabular}

\subsection{Overview of K-stability for Fano varieties}
\label{sec:K_stability_versions}

In the present article we define $K$-semi-stability and uniform $K$-stability using valuations (\autoref{def:K_stability}), which is equivalent then to  the $\delta$-invariant definition (\autoref{def:K_stable}). These definitions were shown to be equivalent in \cite[Theorem B]{BJ} to the more traditional ones that use test configurations. Also, they have a serious disadvantage: there is no known delta invariant type definition of $K$-stability and $K$-polystability, which although we do not use  in any of the statements or in the proofs, they are important notions for the big picture. Hence, for completeness we mention below their definitions using test configurations. We refer the reader to \cite{Tian_Test,Don_Calabi} or more recent papers such as \cite{Dervan_Uniform,Boucksom_Uniform} for more details:
\begin{description}
\item[K-semi-stability] For every normal test configuration, the Donaldson-Futaki invariant is non-negative.
\item[K-stability]  For every normal test configuration, the Donaldson-Futaki invariant is non-negative, and it is equal to zero if and only if the test configuration is a trivial test configuration. In particular, there is no $1$-parameter subgroup of  $\Aut(X)$.
\item[K-poly-stability]  For every normal test configuration the Donaldson-Futaki invariant is non-negative, and it is equal to zero if and only if the test configuration is a product test configuration, i.e. it comes from a one parameter subgroup of the automorphism group of $X$.
\item[Uniform K-stability] There exists a positive real constant $\delta$ such that for every normal test configuration the Donaldson-Futaki invariant is at least $\delta$ times the $L^1$ norm (or, equivalently, the minimum norm) of the test configuration. This notion implies K-stability, and in the case of smooth complex Fanos that the automorphism group of $X$ is finite \cite[Cor E]{Boucksom_Hisamoto_Jonsson_Uniform_K-stability_and_asymptotics_of_energy_functionals_in_Kahler___geometry}.
\end{description}
We also note that the Yau-Tian-Donaldson conjecture asserts that a klt Fano variety admits a singular K\"{a}hler-Einstein metric if and only if it is K-polystable. This is known for smooth \cite{Chen_Donaldson_Sun_Kahler-Einstein_metrics_on_Fano_manifolds_I_Approximation_of_metrics_with_cone_singularities,Chen_Donaldson_Sun_Kahler-Einstein_metrics_on_Fano_manifolds_II_Limits_with_cone_angle_less_than_2pi,Chen_Donaldson_Sun_Kahler-Einstein_metrics_on_Fano_manifolds_III_Limits_as_cone_angle_approaches_2pi_and_completion_of_the_main_proof,Tian} and smoothable Fano varieties \cite{Li_Wang_Xu_Degeneration} (and independently  \cite{SpottiYTD} in the finite automorphism case), and for singular ones admitting a crepant resolution \cite{Recent_Tian}. In the literature, there are also many proposed strenghtening of the notion of K-stability; they should be crucial to extend the YTD conjecture to the case of constant scalar curvature K\"{a}hler metrics. In this paper we are interested in uniform K-stability \cite{Dervan_Uniform,Boucksom_Uniform,Bouck_variational}, which at least for smooth Fano manifold is known to be equivalent to K-stability (we should stress that the proof is via the equivalence with the existence of a  K\"{a}hler-Einstein metric). One can also strenghten the notion of K-stability by looking at possibly non-finitely generated filtration of the coordinate ring, see \cite{Nystrom_filtrations,Gabor,Tits}.

config: arxiv


text:
\begin{document}

\maketitle

\centerline{KATSUKUNI\ \ NAKAGAWA}
\medskip

 \centerline{Graduate School of Advanced Science and Engineering, Hiroshima University}

   \centerline{Kagamiyama 1-3-1, Higashi-Hiroshima 739-8526, Japan}
\centerline{e-mail: ktnakagawa@hiroshima-u.ac.jp}

\vspace{2\baselineskip}
\centerline{\scshape Abstract}
\medskip
\begin{center}
\begin{minipage}{0.8\textwidth}
{\normalsize A decomposition theorem for self-adjoint operators proved by Riesz and Lorch is extended to normal operators.
This extension gives a new proof of the spectral theorem for unbounded normal operators.}
\end{minipage}
\end{center}
\medskip

\section{Introduction}

Let $\HS$ be a Hilbert space.
We denote by $\BH$ the set of all bounded linear operators on $\HS$.
Here is a decomposition theorem for self-adjoint operators proved by Riesz and Lorch in \cite{Riesz-Lorch}.

\begin{thm}[\cite{Riesz-Lorch}]
\label{thm:092}
Let $T:\HS\to\HS$ be a self-adjoint linear operator.
Then there are $A,B\in\BH$ such that the following four assertions hold:
\begin{enumerate}
\item[(i)]$A$ is injective.
\item[(ii)]$T=A^{-1}B$.
\item[(iii)]$AB=BA$.
\item[(iv)]$A$ and $B$ are self-adjoint.
\end{enumerate}
\end{thm}

The first aim of this paper is to extend Theorem \ref{thm:092} to normal operators as follows:

\begin{theorem}
\label{thm:018}
Let $T:\HS\to\HS$ be a normal linear operator.
Then there are $A,B\in\BH$ such that (i)$\sim$(iii) in Theorem \ref{thm:092} and the following assertion hold:
\begin{center}
{\rm (iv)} $A$ is self-adjoint and $B$ is normal.
Moreover, if $T$ is self-adjoint, then so is $B$.
\end{center}
\end{theorem}

To explain the second aim of this paper, we recall the next spectral theorem for normal operators:

\begin{thm}
\label{thm:065}
Let $T:\HS\to\HS$ be a normal linear operator.
Then there is a Borel spectral measure $E$ such that the following equality holds:
\[
T=\int_{\C}z\,E(dz).
\]
\end{thm}

In \cite{Riesz-Lorch}, Riesz and Lorch proved the spectral theorem for self-adjoint operators.
(Their proof is extend in \cite{Conway} and \cite{Rudin} to normal operators.)
They first established the result for bounded self-adjoint operators, then considered the unbounded case.
Theorem \ref{thm:092} was used to derive the result for the unbounded case from that for the bounded one.
In their argument, the commutativity $AB=BA$ is used a bit indirectly.
Moreover, for some reason or other, the self-adjointness of $B$ is not used.

The second aim of this paper is to give an argument with a direct use of the commutativity $AB=BA$ and the normality of $B$.
To this end, the product of two spectral measures is essential.
Our argument gives a natural way to derive Theorem \ref{thm:065} from the bounded version of it.

This paper is organized as follows.
In Section \ref{sec:decomposition}, we prove Theorem \ref{thm:018}.
In Section \ref{sec:spec thm}, we first introduce product spectral measures, then we prove Theorem \ref{thm:065}, using Theorem \ref{thm:018}.

There are various proofs of the spectral theorem.
For a historical overview of this theorem and its proof, see \cite[Section X.9]{Dunford-Schwartz}.

\section{Proof of Theorem \ref{thm:018}}
\label{sec:decomposition}

Let $T:\HS\to\HS$ be a linear operator.
We denote by $\D(T)$ and $\RA(T)$ the domain and the range of $T$, respectively.
For two operators $T$ and $S$, we write $T\subset S$ if $\D(T)\subset\D(S)$ and $Tx=Sx$ holds for all $x\in\D(T)$.
If $T\subset S$ and $T\supset S$, then we write $T=S$.
We say that $T$ is \emph{densely defined} if $\D(T)$ is dense in $\HS$.
For a densely defined operator $T$, we denote by $T^{*}$ the adjoint operator of $T$.
We recall the definitions of a normal operator and a self-adjoint operator.

\begin{definition}
Let $T:\HS\to\HS$ be a densely defined operator.
We say that $T$ is \emph{normal} if $T^*T=TT^*$.
Moreover, we say that $T$ is \emph{self-adjoint} if $T^*=T$.
\end{definition}

Let $I:\HS\to\HS$ be the identity mapping.
For the proof of Theorem \ref{thm:018}, we need the following theorem:

\begin{theorem}
\label{thm:500}
Let $T:\HS\to\HS$ be a closed and densely defined operator.
Then the following three assertions hold:
\begin{enumerate}
\item[(i)]$T^{*}T:\HS\to\HS$ is self-adjoint.
\item[(ii)]$I+T^{*}T$ is injective and $\RA(I+T^{*}T)=\HS$.
\item[(iii)]$(I+T^{*}T)^{-1}$ belongs to $\BH$ and is self-adjoint.
\end{enumerate}
\end{theorem}
\begin{proof}
See, e.g., \cite[Proposition X.4.2]{Conway}.
\end{proof}

Let us prove Theorem \ref{thm:018}.

\begin{proof}[Proof of Theorem \ref{thm:018}]
Thanks to Theorem \ref{thm:500}, we can define $A,B:\HS\to\HS$ by
\begin{align*}
A=(I+T^{*}T)^{-1},\quad B=TA.
\end{align*}
Then, $A\in\BH$ from Theorem \ref{thm:500} (iii).
We show $B\in \BH$.
Since $\RA(A)=\D(I+T^{*}T)=\D(T^{*}T)\subset\D(T)$, we have $\D(B)=\D(A)=\HS$.
Moreover, $B$ is closed since $T$ is closed and $A$ is bounded.
Therefore, by the closed graph theorem, we have $B\in\BH$.

(i) $A$ is injective from Theorem \ref{thm:500} (ii).

(ii)\ We first show the following claim:

\begin{cla}
\label{cla:012}
\normalfont
$AT\subset B$.
\end{cla}
\begin{proof}[Proof of Claim \ref{cla:012}]
We have $\D(AT)\subset\D(B)$ since $\D(B)=\HS$.
Let $x\in\D(AT)$.
Theorem \ref{thm:500} (ii) implies that there is $y\in\D(I+T^{*}T)$ such that $x=(I+T^{*}T)y$.
Clearly, $Bx=Ty$.
On the other hand, since $T$ is normal, $T(I+T^{*}T)=T+(TT^{*})T=T+(T^{*}T)T=(I+T^{*}T)T$, and hence, $ATx=(I+T^{*}T)^{-1}((I+T^{*}T)Ty)=Ty$.
Combining, we obtain $ATx=Bx$.
\end{proof}

We prove (ii).
By Claim \ref{cla:012}, we have $T=(A^{-1}A)T=A^{-1}(AT)\subset A^{-1}B$.
It remains to show $\D(T)\supset\D(A^{-1}B)$.
Let $x\in\D(A^{-1}B)$.
Then, $Bx\in\D(A^{-1})$.
Theorem \ref{thm:500} (ii) implies that there is $y\in\D(I+T^{*}T)$ such that $x=(I+T^{*}T)y$.
We have $Ty\in\D(A^{-1})$ since $Bx=Ty$.
Moreover, since $T$ is normal, $\D(A^{-1})=\D(I+T^{*}T)=\D(T^{*}T)=\D(TT^{*})$.
Thus, we have $Ty\in\D(TT^{*})$, and hence, $T^{*}Ty\in\D(T)$.
This and $x=y+T^*Ty$ imply $x\in\D(T)$.

(iii)\ It is enough for us to prove $AB\subset BA$.
By Claim \ref{cla:012}, we have $AB=A(TA)=(AT)A\subset BA$.

(iv)\ $A$ is self-adjoint from Theorem \ref{thm:500} (iii).
Notice that $AT$ is densely defined since $\D(A)=\HS$ and  $T$ is densely defined.
Thus, we can define $(AT)^{*}$.
Moreover, by Claim \ref{cla:012}, we have $B^{*}\subset(AT)^{*}=T^{*}A$. This and $\D(B^{*})=\HS$ imply
\begin{equation}
\label{eq:806}
B^{*}=T^{*}A.
\end{equation}
By (\ref{eq:806}) and the commutativity (iii), we have $B^{*}B=T^{*}AB=T^{*}BA=(T^{*}T)AA$.
Since $T$ is normal, $(T^{*}T)AA=(TT^{*})AA$.
Moreover, (iii) and the self-adjointness of $A$ imply $AB^{*}=B^{*}A$, and hence, $(TT^{*})AA=TB^{*}A=TAB^{*}=BB^{*}$.
Combining, we conclude that $B^{*}B=BB^{*}$, that is, $B$ is normal.
Moreover, if $T$ is self-adjoint, then $B^{*}=TA=B$ from (\ref{eq:806}), and hence, we conclude that $B$ is self-adjoint.
\end{proof}

\section{Proof of the spectral theorem}
\label{sec:spec thm}

We quote the spectral theorem for bounded normal operators:
\begin{thm}
\label{thm:073}
Let $T:\HS\to\HS$ be a normal operator.
If $T$ is bounded, then there is a Borel spectral measure $E$ such that the following equality holds:
\[
T=\int_{\C}z\,E(dz).
\]
\end{thm}
For the proof of Theorem \ref{thm:073}, see, e.g., \cite[Theorem IX.2.2]{Conway}.

The aim of this section is to derive Theorem \ref{thm:065} from Theorem \ref{thm:073} via Theorem \ref{thm:018}, with a direct use of the commutativity $AB=BA$ and the normality of $B$.
To this end, we need the following theorem:

\begin{theorem}
\label{thm:022}
Let $\Omega_1$ and $\Omega_2$ be locally compact Hausdorff topological spaces with countable open basis and let $E_1$ and $E_2$ be Borel spectral measures on $\Omega_1$ and $\Omega_2$, respectively.
Assume that $E_1$ and $E_2$ act on the same Hilbert space and commute, that is, $E_1(A)E_2(B)=E_2(B)E_1(A)$ holds for any Borel sets $A\subset\Omega_1$ and $B\subset\Omega_2$.
Then there is a unique Borel spectral measure $E_1\times E_2$ on $\Omega_1\times\Omega_2$ such that
\[
(E_1\times E_2)(A\times B)=E_1(A)E_2(B)
\]
holds for any Borel sets $A\subset\Omega_1$ and $B\subset\Omega_2$.
\end{theorem}
\begin{proof}
See, e.g., \cite[Theorem 4.10]{Schmudgen}.
\end{proof}

We call the above $E_1\times E_2$ the \emph{product spectral measure} of $E_1$ and $E_2$.
We can verify the following Fubini-type theorem:

\begin{theorem}
\label{thm:042}
Let $\Omega_1,\Omega_2$ and $E_1,E_2$ be as in Theorem \ref{thm:022} and let $f_1:\Omega_1\to\C,\,f_2:\Omega_2\to\C$ be Borel measurable functions.
If $f_2$ is bounded, then we have
\[
\left(\int_{\Omega_1}f_1\,dE_1\right)\left(\int_{\Omega_2}f_2\,dE_2\right)=\int_{\Omega_1\times\Omega_2}f_1(\omega_{1})f_2(\omega_{2})\,(E_1\times E_2)(d\omega_{1}d\omega_{2}).
\]
\end{theorem}
\begin{proof}
Let $\pi_1$ and $\pi_2$ be the projections from $\Omega_1\times\Omega_2$ to $\Omega_1$ and $\Omega_2$, respectively.
Then, $\int_{\Omega_i}f_i\,dE_i=\int_{\Omega_1\times\Omega_2}f_i\circ\pi_i\,d(E_1\times E_2)$ holds for $i=1,2$.
Thus, the desired result follows from \cite[Theorem 4.16 (iii)]{Schmudgen}.
\end{proof}

\begin{remark}
For a closable linear operator $T$, we denote by $\overline{T}$ the closure of $T$.
\cite[Theorem 4.16 (iii)]{Schmudgen} actually states that $\overline{(\int f\,dE)(\int g\,dE)}=\int fg\,dE$ holds, where $E$ is a spectral measure on a measurable space $\Omega$, and $f,g:\Omega\to\C$ are measurable functions (they may be unbounded).
In Theorem \ref{thm:042}, by the boundedness of $f_2$, the operator $(\int_{\Omega_1}f_1\,dE_1)(\int_{\Omega_2}f_2\,dE_2)$ is closed, and we need not to take closure.
\end{remark}

We are ready to prove Theorem \ref{thm:065}.

\begin{proof}[Proof of Theorem \ref{thm:065}]
Let $A,B\in \BH$ be as in Theorem \ref{thm:018}.
Since $A$ is self-adjoint and $B$ is normal, from Theorem \ref{thm:073}, there are Borel spectral measures $E_1$ and $E_2$ on $\R$ and $\C$, respectively, satisfying
\[
A=\int_{\R}t\,E_1(dt),\quad B=\int_{\C}w\,E_2(dw).
\]
Moreover, since $B$ is bounded, there is a compact set $K\subset\C$ such that $B=\int_{K}w\,E_2(dw)$.
We define $f:\R\to\R$ by $f(0)=0,\,f(t)=1/t\>(t\neq0)$.
It is easy to show
\[
A^{-1}=\int_{\R}f(t)\,E_1(dt).
\]
The commutativity $AB=BA$ implies that $E_1$ and $E_2$ commute.
Therefore, from Theorem \ref{thm:022}, the product spectral measure $E_1\times E_2$ on $\R\times K$ exists.
By Theorem \ref{thm:042}, we have
\[
T=\int_{\R\times K}f(t)w\,(E_1\times E_2)(dt\,dw)=\int_{\C}z\,E(dz),
\]
where $E$ denotes the push forward of $E_1\times E_2$ by the map $\R\times K\ni(t,w)\mapsto f(t)w\in\C$.
We obtain the desired result.
\end{proof}

\centerline{\scshape Acknowledgments}
\medskip
The author would like to thank Mishio Kawashita (Hiroshima University), Koichiro Iwata (Hiroshima University) and Masao Hirokawa (Kyushu University) for helpful advices and comments.
The author is supported by FY2019 Hiroshima University Grant-in-Aid for Exploratory Research (The researcher support of young Scientists).


TITLE: A group acting on colourings of a set
QUESTION [6 upvotes]: Suppose I have a set $L$ with some permutation group $G$ defined upon it, which I think of as a symmetry group. I want to consider the set $F$ of functions $f: L \to C$, for some set $C$. It seems that there should be a natural way in which the group $G$ can also be applied to this set of functions, but I'm not sure of the best way to think about it in terms of abstract algebra.
As a concrete example, let $L$ be the faces of a cube, and let $G$ be the group of rotations of the cube. Then $F$ is equivalent to the set of colourings of the cube's faces, with $C$ being the set of possible colours. Given such a colouring, the cube can be rotated to produce another colouring. This allows us, for example, to define a set of equivalence classes where two colourings are considered equivalent if the cube can be rotated to produce one from the other.
It seems there at least two ways one could think about this. One is to define a group action of $G$ upon $F$. Another is to think about the symmetries of $F$ as being a group in its own right, which is somehow derived from $L$ and $|C|$. Perhaps there are others. But in either case I don't quite know how to express the idea formally.
This seems like it should be a fairly common concept with a collection of well-known standard results, and perhaps some nice generalisations. However, I don't know its name, and the introductory group theory texts I've consulted don't seem to cover it, so I would be grateful if anyone can point me in the right direction.
My real reason for wanting to do this is as follows: a cellular automaton is a type of dynamical system, whose state space is the set of colourings of an infinite lattice. The state transition function is typically invariant to translations of the lattice, and optionally also to rotations and reflections. I would like to generalise this concept, allowing us to talk about cellular automata as discrete dynamical systems with a particular type of symmetry.

REPLY [8 votes]: $G$ acts on functions by $f(x) \mapsto f(g^{-1} x)$. This kind of action is studied, for example, in combinatorics in the context of the Pólya enumeration theorem. It is also studied in harmonic analysis, with $C$ the real or complex numbers, where it is used to generalize the Fourier transform; see Pontryagin duality.
More abstractly, the construction $L \mapsto \text{Hom}(L, C)$, where $\text{Hom}$ denotes the set of functions from $L$ to $C$, is an example of a contravariant functor. In particular, that means if a monoid $M$ acts on $L$, then the opposite monoid $M^{op}$ naturally acts on functions $L \to C$ via $f(x) \mapsto f(mx)$. In the special case of groups inversion provides a canonical isomorphism $G \cong G^{op}$, so we can turn an action of $G^{op}$ into an action of $G$. This is done in, for example, representation theory in order to construct dual representations.


text:
\begin{document}

\begin{center}
{\LARGE Asymptotic Properties of Self-Normalized Linear Processes with Long
Memory }\vskip15pt

\bigskip Magda Peligrad$^{a}$\footnote{Supported in part by a Charles Phelps
Taft Memorial Fund grant and NSA\ grant H98230-09-1-0005.} and Hailin
Sang$^{b}$
\end{center}

$^{a}$ Department of Mathematical Sciences, University of Cincinnati, PO Box
210025, Cincinnati, OH 45221-0025, USA. E-mail address: peligrm@ucmail.uc.edu

\bigskip

$^{b}$ National Institute of Statistical Sciences, PO Box 14006, Research
Triangle Park, NC 27709, USA. E-mail address: sang@niss.org

\begin{center}
\bigskip
\end{center}

\textit{Mathematical Subject Classification} (2010): 60F05, 60F17, 60G10, 60G22.

Key words and phrases: linear processes, long memory, invariance principle,
fractional Brownian motion, domain of normal attraction, fractionally
integrated processes, unit root.

\bigskip

\textbf{Abstract}

\bigskip

In this paper we study the convergence to fractional Brownian motion for long
memory time series having independent innovations with infinite second moment.
For the sake of applications we derive the self-normalized version of this
theorem. The study is motivated by models arising in economical applications
where often the linear processes have long memory, and the innovations have
heavy tails.

\section{Introduction and notations}

In this paper we study the asymptotic properties of a causal linear process
\begin{equation}
X_{k}=\sum_{i\geq0}a_{i}\varepsilon_{k-i}\label{X}
\end{equation}
when the i.i.d. innovations $\{\varepsilon,$ $\varepsilon_{n};n\in
\mathbb{Z}\}$ have infinite variance and $\{a_{i};i\geq0\}$ is a sequence of
real constants such that $X_{k}$ is well defined. More precisely, everywhere
in the paper, we assume that the innovations are centered and in the domain of
attraction of a normal law. This means that the variables are independent,
identically distributed,
\begin{equation}
E\varepsilon=0\label{centered}
\end{equation}
and
\begin{equation}
l(x)=E\varepsilon^{2}I(|\varepsilon|\leq x)\text{ is a slowly varying function
at }\infty\text{ .}\label{DA}
\end{equation}
We say that $h(t),$ defined for $t\geq0$, is slowly varying if it is positive
and measurable on $[A,\infty),$ for some $A>0,$ and if for any $\lambda>0$, we
have $\lim_{x\rightarrow\infty}h(\lambda x)/h(x)=1$ (Seneta, 1970, Definition 1.1).

We define
\[
S_{n}=\sum_{i=1}^{n}X_{i}\text{ .}
\]
The central limit theorem for $S_{n}$ with i.i.d. innovations and infinite
variance when $\sum_{i\geq0}|a_{i}|<\infty\ $ was studied by many authors. We
mention among them, Knight (1991), Mikosch et al (1995), Wu (2003). For this
case the central limit theorem was obtained under a normalization that is
regularly varying with exponent $1/2.$

The purpose of this paper is to investigate the central limit theorem in its
functional form for the case when
\begin{equation}
a_{n}=n^{-\alpha}L(n),\text{ where }1/2<\alpha<1, n\geq1 \label{dcoeff}
\end{equation}
and $L(n)$ is a slowly varying function at $\infty$ in the strong sense (i.e.
there is a slowly varying function $h(t)$ such that $L(n)=h(n)$). Notice that,
by the definition of slowly varying function, the coefficients $a_{n}$ are
positive for $n$ sufficiently large. We shall obtain convergence in
distribution under a normalization that is regularly varying with exponent
$3/2-\alpha$ which is strictly larger than $1/2.$ This is the reason why the
time series we consider has long memory.

To give an example of a linear process of this type we mention the
fractionally integrated processes since they play an important role in
financial time series modeling and they are widely studied. Such processes are
defined for $0<d<1/2$ by
\begin{equation}
X_{k}=(1-B)^{-d}\varepsilon_{k}=\sum_{i\geq0}a_{i}\varepsilon_{k-i}\text{ with
}a_{i}=\frac{\Gamma(i+d)}{\Gamma(d)\Gamma(i+1)}\,\text{\ ,}
\label{deffractlin}
\end{equation}
where $B$ is the backward shift operator, $B\varepsilon_{k}=\varepsilon_{k-1}
$. For this example, by the well known fact that for any real $x,$
$\lim_{n\rightarrow\infty}\Gamma(n+x)/n^{x}\Gamma(n)=1,$ we have$\ \lim
_{n\rightarrow\infty}a_{n}/n^{d-1}=1/\Gamma(d)$.

The CLT\ in its functional form was intensively studied for the case of i.i.d.
innovations with finite second moment. We refer to Davydov (1970), Taqqu
(1975), Phillips and Solo (1992), Wang et al (2003), Wu and Min (2005),
Dedecker et al (2009), among others. Invariance principles (or functional
central limit theorems) play an important role in econometrics and statistics.
For example, to obtain asymptotic distributions of unit-root test statistics,
researchers have applied invariance principles of various forms; see Phillips
(1987) and Wu (2006).

We shall derive here the central limit theorem and its functional form, i.e.
convergence to fractional Brownian motion, for the case when the innovations
are in the domain of attraction of the normal distribution and the constants
satisfy (\ref{dcoeff}). The normalizer in this theorem depends on the slowly
varying function $l(x)$ that is in general unknown. To make our results easily
applicable we also study the central limit theorem in its self-normalized form.

The self-normalized CLT\ for sums of independent identically distributed
random variables was treated in the paper by Gin\'{e} et al (1997). The case
of self-normalized sums in the domain of attraction of other stable laws was
considered by Chistyakov and G\"{o}tze (2004). A systematic treatment of
self-normalized limit theory under independence assumption is given in the
book by de la Pe\~{n}a et al (2009). The self-normalized version of the
functional central limit theorem for this case, was treated in Cs\"{o}rg\H{o}
et al (2003). Kulik (2006) studied the self-normalized functional CLT\ when
$\sum_{i\geq0}|a_{i}|<\infty.$ We shall consider the long memory case when
coefficients satisfy (\ref{dcoeff}).

Our paper is organized in the following way: Section 2 contains the
definitions and the results; the proofs are given in section 3. For
convenience, in the Appendix, we give some auxiliary results and we also
mention some known facts needed for the proofs.

In this paper we shall use the following notations: a double indexed sequence
with indexes $n$ and $i$ will be denoted by $a_{ni}$ when no confusion is
possible, and sometimes by $a_{n,i};$ we use the notation $a_{n}\mathbb{\sim
}b_{n}$ instead of $a_{n}/b_{n}\rightarrow1$; for positive sequences, the
notation $a_{n}\ll b_{n}$ replaces Vinogradov symbol $O$ and it means that
$a_{n}/b_{n}$ is bounded; $a_{n}=o(b_{n})$ stays for $a_{n}/b_{n}
\rightarrow0;$ $[x]$ denotes the integer part of $x$; the notation
$\Rightarrow$ is used for weak convergence, and $\overset{P}{\rightarrow}$
denotes convergence in probability. By $var(X)$ we denote the variance of the
random variable $X$ and by $cov(X,Y)$ the covariance of $X$ and $Y$. The weak
convergence to a constant means convergence in probability. We denote by
$D[0,1]$ the space of all functions on $[0,1]$ which have left-hand limits and
are continuous from the right. $N(0,1)$ denotes a standard normal random variable.

\section{Results}

To introduce our results we define a normalizing sequence in the following
way. Recall (\ref{DA}) and (\ref{dcoeff}). Let $b=\inf\left\{  x\geq
1:l(x)>0\right\}  $, define
\begin{equation}
\eta_{j}=\inf\left\{  s:s\geq b+1,\frac{l(s)}{s^{2}}\leq\frac{1}{j}\right\}
,\;\;\;j=1,2,\cdots\label{defeta}
\end{equation}
and set
\begin{equation}
B_{n}^{2}:=c_{\alpha}l_{n}n^{3-2\alpha}L^{2}(n)\text{ with }l_{n}=l(\eta_{n})
\label{defBn}
\end{equation}
where
\begin{equation}
c_{\alpha}=\{\int_{0}^{\infty}[x^{1-\alpha}-\max(x-1,0)^{1-\alpha}
]^{2}dx\}/(1-\alpha)^{2}\text{ .} \label{defcalpha}
\end{equation}

\begin{theorem}
\label{clt} Define $\{X_{n};n\geq1\}$ by (\ref{X}) and the random element
$W_{n}(t)=S_{[nt]}/B_{n}$ on the space $D[0,1]$. Assume conditions
(\ref{centered}), (\ref{DA}) and (\ref{dcoeff}) are satisfied. Then,
$W_{n}(t)$ converges weakly on the space $D[0,1]$ endowed with Skorohod
topology to the fractional Brownian motion $W_{H}$ with Hurst index
$H=3/2-\alpha$. \newline In particular, for $t=1,$ we have that $S_{n}/B_{n}$
converges in distribution to a standard normal variable.
\end{theorem}

\begin{remark}
In a forthcoming paper the authors treat the central limit theorem for the
situation when $B_{n}^{2}$ is not necessarily regularly varying. However, for
that situation the convergence to the fractional Brownian motion might fail.
As a matter of fact, in the context of Theorem \ref{clt} a necessary condition
for the convergence to the fractional Brownian motion $W_{H}$ with Hurst index
$H=\beta$ is the representation $B_{n}^{2}=n^{2\beta}h(x)$ for a function
$h(x)$ that is slowly varying at infinity (see Lamperti, 1962).
\end{remark}

For successfully applying this theorem we have to know $l_{n}$ that depends on
the distribution of $\varepsilon.$ This can be avoided by constructing a
selfnormalizer. Denote $\sum_{i=0}^{\infty}a_{i}^{2}=A^{2}.$ Our result is:

\begin{theorem}
\label{selfCLT}Under the same conditions as in Theorem \ref{clt} we have
\begin{equation}
\frac{1}{nl_{n}}\sum_{i=1}^{n}X_{i}^{2}\overset{P}{\rightarrow}A^{2}
\label{LLN}
\end{equation}
and therefore
\[
\frac{S_{[nt]}}{na_{n}\sqrt{\sum_{i=1}^{n}X_{i}^{2}}}\Rightarrow\frac
{\sqrt{c_{\alpha}}}{A}W_{H}(t)\text{ .}
\]
In particular
\[
\frac{S_{n}}{na_{n}\sqrt{\sum_{i=1}^{n}X_{i}^{2}}}\Rightarrow N(0,\frac
{c_{\alpha}}{A^{2}})\text{ .}
\]

\end{theorem}

\section{Application to unit root testing}

Invariance principles play an important role in characterizing the limit
distribution of various statistics arising from the inference in economic time series.

Let us consider a stochastic process generated according to

\[
Y_{n}=\rho Y_{n-1}+X_{n}\text{ for }n\geq1
\]
where $Y_{0}=0$ and $(X_{n})_{n\geq1}$ is a stationary sequence and $\rho$ is
a constant. Denote the ordinary least squares (OLS) estimator of $\rho$ by
\[
\hat{\rho}_{n}=\sum_{k=1}^{n}Y_{k}Y_{k-1}/\sum_{k=1}^{n}Y_{k-1}^{2}\text{ .}
\]
To test $\rho=1$ against $\rho<1$, a key step is to derive the limit
distribution of the well-known Dickey--Fuller (DF) test statistic (Dickey and
Fuller 1979, 1981):

\[
\hat{\rho}_{n}-1=\sum_{k=1}^{n}Y_{k-1}(Y_{k}-Y_{k-1})/\sum_{k=1}^{n}
Y_{k-1}^{2}\text{ .}
\]
As shown by Phillips (1987), under the null hypothesis $\rho=1$, the
asymptotic properties of the DF test statistic rely heavily on the invariance
principles. This problem was widely studied under various assumptions on the
sequence $X_{n}.$ Among them Sowell (1990) and Wu (2006) considered the unit
root testing problem for long-memory processes. By combining our Theorems
\ref{clt} and \ref{selfCLT} with arguments similar to Phillips (1987), we can
formulate the following result obtained for variables that do not necessarily
have finite second moment.

\begin{proposition}
Assume that $(X_{n})_{n\geq1}$ is as in Theorem \ref{clt}. Then the following
results hold
\[
(a)\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\frac{\sum_{k=1}^{n}
Y_{k-1}^{2}}{n^{3}a_{n}^{2}\sum_{i=1}^{n}X_{i}^{2}}\Rightarrow\text{\ }
\frac{c_{\alpha}}{A^{2}}\int_{0}^{1}W_{H}^{2}(t)dt\text{ .\ }
\]
\[
(b)\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\frac{\sum_{k=1}^{n}
Y_{k-1}(Y_{k}-Y_{k-1})}{n^{2}a_{n}^{2}\sum_{i=1}^{n}X_{i}^{2}}\Rightarrow
\frac{c_{\alpha}W_{H}^{2}(1)}{2A^{2}}\text{ .}
\]
\[
(c)\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }n(\hat{\rho}_{n}-1)\Rightarrow
\frac{W_{H}^{2}(1)/2}{\int_{0}^{1}W_{H}^{2}(t)dt}\text{ }.\text{
\ \ \ \ \ \ \ \ \ \ \ \ }
\]

\end{proposition}

The proof of this proposition requires only to make obvious changes in the
proofs of (A1) and (A2) on page 296 in Phillips (1987), and it is left to the reader.

\section{Proofs}

\subsection{Proof of theorem \ref{clt}}

In order to prove the central limit theorem in its functional form,\textbf{
}i.e. the weak convergence of $S_{[nt]}/B_{n}$ on the space $D[0,1]$ to the
fractional Brownian motion $W_{H}$ with Hurst index $H=3/2-\alpha$, we shall
first reduce the problem to truncated random variables. For the truncated
process we establish tightness on $D[0,1]$ and the convergence of finite
dimensional distributions.

Without the loss of generality, in the rest of the paper, we assume for
convenience $a_{0}=0$ in definition (\ref{X}).

We shall divide the proof in several steps:

\textbf{Step 1. Existence.}

To show that $X_{1}$ is well defined we use stationarity and Lemma
\ref{csorgo} from the Appendix. First of all we have
\[
\sum_{i=1}^{\infty}P(|a_{i}\varepsilon_{1-i}|>1)=\sum_{i=1}^{\infty
}P(|\varepsilon|>|a_{i}|^{-1})=\sum_{i=1}^{\infty}a_{i}^{2}o(l(|a_{i}
|^{-1}))\text{ .}
\]
Then, by taking into account that (\ref{centered}) implies $E\varepsilon
I(|\varepsilon|\leq|a_{i}|^{-1})=-E\varepsilon I(|\varepsilon|>|a_{i}
|^{-1}),$
\[
\sum_{i=1}^{\infty}|Ea_{i}\varepsilon_{1-i}I(|a_{i}\varepsilon_{1-i}
|\leq1)|\leq\sum_{i=1}^{\infty}|a_{i}|E|\varepsilon|I(|\varepsilon
|>|a_{i}|^{-1})=\sum_{i=1}^{\infty}a_{i}^{2}o(l(|a_{i}|^{-1}))\text{ }
\]
and
\[
\sum_{i=1}^{\infty}Ea_{i}^{2}\varepsilon_{1-i}^{2}I(|a_{i}\varepsilon
_{1-i}|\leq1)=\sum_{i=1}^{\infty}a_{i}^{2}E\varepsilon^{2}I(|\varepsilon
|\leq|a_{i}|^{-1})=\sum_{i=1}^{\infty}a_{i}^{2}l(|a_{i}|^{-1})\text{ .}
\]
Notice that
\[
\sum_{i=1}^{\infty}a_{i}^{2}l(|a_{i}|^{-1})=\sum_{i=1}^{\infty}i^{-2\alpha
}L^{2}(i)l(i^{2\alpha}L^{-2}(i))<\infty\text{ ,}
\]
since $1/2<\alpha<1$ and $L^{2}(i)l(i^{2\alpha}L^{-2}(i))$ is a slowly varying
function at $\infty$. The existence in the almost sure sense follows by
combining these arguments with the three series theorem.

\textbf{Step 2}. \textbf{Truncation.}

For the case when$~E\varepsilon^{2}=\infty$, which is relevant to our paper,
the truncation is necessary. The challenge is to find a suitable level of
truncation. For any integer $1\leq k\leq n$ define
\begin{equation}
X_{nk}^{\prime}=\sum_{i=1}^{\infty}a_{i}\varepsilon_{k-i}I(|\varepsilon
_{k-i}|\leq\eta_{n-k+i})\text{ and }S_{n}^{\prime}=\sum_{k=1}^{n}
X_{nk}^{\prime}\text{ .} \label{defprime}
\end{equation}
This definition has the advantage that $S_{n}^{\prime}$ can be expressed as a
simple sum of a linear process of an array of independent variables. For every
$m\geq1$ we denote
\begin{equation}
b_{m}=a_{1}+\ldots+a_{m}\text{ }, \label{defbm}
\end{equation}
and then we introduce the coefficients
\begin{align}
b_{nj}  &  =b_{j}=a_{1}+\ldots+a_{j}\text{ for }j<n\label{defbn}\\
b_{nj}  &  =b_{j}-b_{j-n}=a_{j-n+1}+\ldots+a_{j}\,\text{\ for }j\geq n\text{
.}\nonumber
\end{align}
With this notation and recalling definition (\ref{defeta}), by changing the
order of summation,
\begin{equation}
S_{n}^{\prime}=\sum_{i\geq1}b_{ni}\varepsilon_{n-i}I(|\varepsilon_{n-i}
|\leq\eta_{i})\text{ .} \label{repS}
\end{equation}
We shall reduce next, the study of limiting distribution of $S_{n}/B_{n}$ to
the sequence $S_{n}^{\prime}/B_{n}.$ It is enough to show that
\begin{equation}
\frac{1}{B_{n}}E|S_{n}-S_{n}^{^{\prime}}|\rightarrow0\text{ .} \label{L1}
\end{equation}
To see this we use the fact that by Lemma \ref{csorgo} stated in the Appendix
\[
E|\varepsilon|I(|\varepsilon|>\eta_{i})=o(\eta_{i}^{-1}l_{i})\text{ .}
\]
We also know that
\begin{equation}
\eta_{n}^{2}\sim nl_{n}\text{ .} \label{ordereta}
\end{equation}
(see for instance relation 13 in Cs\"{o}rg\H{o} et al, 2003).\ Then, by the
triangle inequality and relation (\ref{boundb}) of Lemma \ref{coeff} from the
Appendix applied with $p=1$, we obtain
\begin{gather}
E|S_{n}-S_{n}^{^{\prime}}|\leq\sum_{i\geq1}|b_{ni}|E|\varepsilon
|I(|\varepsilon|>\eta_{i})=\sum_{i\geq1}|b_{ni}|o(\eta_{i}^{-1}l_{i}
)\label{negl2}\\
=\sum_{i\geq1}|b_{ni}|o(i^{-1/2}l_{i}^{1/2})=o(n^{3/2-\alpha}l_{n}
^{1/2}L(n))=o(B_{n})\nonumber
\end{gather}
and so (\ref{L1}) is established.

\textbf{Step} \textbf{3.} \textbf{Central limit theorem.}

To make the proof more transparent we shall present first the central limit
theorem for $S_{n}/B_{n}$. By the Step 2 it is enough to find the limiting
distribution of $S_{n}^{^{\prime}}/B_{n}$. We start by noticing that by
(\ref{negl2}) and the fact that the variables are centered we have
\begin{equation}
|ES_{n}^{^{\prime}}|=|E(S_{n}-S_{n}^{^{\prime}})|=o(B_{n})\text{ .}
\label{negl5}
\end{equation}
One of the consequences of this observation is that $S_{n}^{^{\prime}}/B_{n}$
has the same limiting distribution as $(S_{n}^{^{\prime}}-ES_{n}^{\prime
})/B_{n}$. Furthermore,
\[
var(\frac{S_{n}^{^{\prime}}}{B_{n}})=\frac{1}{B_{n}^{2}}\sum_{i\geq1}
b_{ni}^{2}l_{i}-\frac{1}{B_{n}^{2}}(ES_{n}^{^{\prime}})^{2}\rightarrow1
\]
by relation (\ref{VAR}) in Lemma \ref{coeff} and (\ref{negl5}).

Moreover, by the point 1 in Lemma \ref{coeff} for $k_{n}=n^{4/(2\alpha-1)}$
\begin{align}
var(\sum_{i\geq k_{n}}b_{ni}\varepsilon_{n-i}I(|\varepsilon_{n-i}|  &
\leq\eta_{i}))\ll\sum_{i\geq k_{n}}n^{2}(i-n)^{-2\alpha}L^{2}(i)l_{i}
\label{n}\\
&  =o(1)\text{ as }n\rightarrow\infty\text{ .}\nonumber
\end{align}
Then, by Theorem 4.1 in Billingsley (1968), for proving the central limit
theorem it is enough to verify Lyapunov's condition for $B_{n}^{-1}(\bar
{S}_{n}^{\prime}-E\bar{S}_{n}^{\prime})$ where
\[
\bar{S}_{n}^{\prime}=\sum_{i=1}^{k_{n}}b_{ni}\varepsilon_{n-i}I(|\varepsilon
_{n-i}|\leq\eta_{i})\text{ .}
\]
Clearly, by (\ref{n}), $var(\bar{S}_{n}^{\prime}/B_{n})\rightarrow1$. In the
estimate below we use the point 4 of Lemma \ref{csorgo} along with
(\ref{ordereta}), followed by relation (\ref{boundb}) of Lemma \ref{coeff}
applied with $p=3$ and the fact that $B_{n}\rightarrow\infty$ to get:
\begin{gather}
\sum_{j=1}^{k_{n}}|b_{nj}|^{3}E|\varepsilon^{\prime}-E\varepsilon^{\prime
}|^{3}\leq8\sum_{j=1}^{k_{n}}|b_{nj}|^{3}E|\varepsilon|^{3}I(|\varepsilon
|\leq\eta_{j})\label{Leap}\\
=\sum_{j=1}^{k_{n}}|b_{nj}|^{3}\eta_{j}o(l_{j})\leq\sum_{j=1}^{\infty}
|b_{nj}|^{3}j^{1/2}o(l_{j}^{3/2})=o(n^{3(3/2-\alpha)}l_{n}^{3/2}
L^{3}(n))=o(B_{n}^{3})\text{ .}\nonumber
\end{gather}
By Lyapunov's central limit theorem and the above considerations, $S_{n}
/B_{n}$ converges to $N(0,1)$ in distribution.

\textbf{Step 4. Preliminary considerations for the convergence to fractional
Brownian motion.}

For $n\geq1$ fixed we implement the same level of truncation as before and
construct $\{X_{nj}^{\prime};1\leq j\leq n\}$ by definition (\ref{defprime}).
Then we introduce the processes
\[
W_{n}^{\prime}(t)=\frac{1}{B_{n}}\sum_{j=1}^{[nt]}X_{nj}^{\prime}\text{ and
}W_{n}^{^{\prime\prime}}(t)=W_{n}(t)-W_{n}^{\prime}(t)\text{ .}
\]
We shall show first that $W_{n}^{^{\prime\prime}}(t)$ is negligible for the
weak convergence on $D[0,1]$ and then, in the next steps, that $W_{n}^{\prime
}(t)$ is weakly convergent to the fractional Brownian motion.

In order to explain this step, it is convenient to express the process in an
expanded form. By using notation (\ref{defbm})
\[
W_{n}^{^{\prime\prime}}(t)=\frac{1}{B_{n}}\sum_{i=0}^{[nt]-1}b_{[nt]-i}
\varepsilon_{i}I(|\varepsilon_{i}|>\eta_{n-i})+\frac{1}{B_{n}}\sum_{i\geq
1}(b_{[nt]+i}-b_{i})\varepsilon_{-i}I(|\varepsilon_{-i}|>\eta_{n+i})\text{ .}
\]
We notice that by the triangle inequality,
\begin{gather*}
E(\sup_{0\leq t\leq1}|W_{n}^{^{\prime\prime}}(t)|)\leq\frac{1}{B_{n}}
E(\sup_{0\leq t\leq1}|\sum_{i=0}^{[nt]-1}b_{[nt]-i}\varepsilon_{i}
I(|\varepsilon_{i}|>\eta_{n-i})|)\\
+\frac{1}{B_{n}}E(\sup_{0\leq t\leq1}|\sum_{i\geq1}(b_{[nt]+i}-b_{i}
)\varepsilon_{-i}I(|\varepsilon_{-i}|>\eta_{n+i})|)\text{ .}
\end{gather*}
Then, by monotonicity and using the notation (\ref{defbn})
\begin{gather}
E(\sup_{0\leq t\leq1}|W_{n}^{^{\prime\prime}}(t)|)\leq\frac{1}{B_{n}}
\sum_{i=0}^{n-1}|b_{n-i}|E|\varepsilon|I(|\varepsilon|>\eta_{n-i}
)\label{negl3}\\
+\frac{1}{B_{n}}\sum_{i\geq1}|b_{n+i}-b_{i}|E|\varepsilon|I(|\varepsilon
|>\eta_{n+i})=\frac{1}{B_{n}}\sum_{i\geq1}|b_{ni}|E|\varepsilon|I(|\varepsilon
|>\eta_{i})\text{ .}\nonumber
\end{gather}
which is exactly the quantity shown to converge to $0$ in (\ref{negl2}). By
Theorem 4.1 in Billingsley (1968), it is enough to study the limiting behavior
of $W_{n}^{\prime}(t)$.

\textbf{Step 5. Tightness.}

As before, we reduce the problem to studying the same problem for
$W_{n}^{\prime}(t)-EW_{n}^{\prime}(t).$ This is easy to see, since, by the
fact the variables are centered and by (\ref{negl3}) we clearly obtain
\begin{equation}
\sup_{0\leq t\leq1}|EW_{n}^{\prime}(t)|=\sup_{0\leq t\leq1}|EW_{n}
^{^{\prime\prime}}(t)|\leq E(\sup_{0\leq t\leq1}|W_{n}^{^{\prime\prime}
}(t)|)\rightarrow0\text{ .} \label{negl4}
\end{equation}
In order to show that $W_{n}^{\prime}(t)-EW_{n}^{\prime}(t)$ is tight in
$D[0,1]$ we shall verify the conditions from Lemma \ref{Ltight}, in Appendix,
for the triangular array $B_{n}^{-1}(X_{nk}^{\prime}-EX_{nk}^{\prime})$,
$1\leq k\leq n.$ This will be achieved in the following two lemmas.

By the properties of slowly varying functions (see Seneta 1976 and also Lemma
\ref{Karamata} in Appendix) we construct first an integer $N_{0}$ and positive
constants $K_{i}$ such that for all $m>N_{0}$ we have simultaneously
\begin{equation}
\max_{1\leq j\leq m}b_{j}^{2}\leq K_{1}m^{2-2\alpha}L^{2}(m)\text{ ,}
\label{1}
\end{equation}
\begin{equation}
l_{2m}\leq K_{2}l_{m}\text{ ,} \label{2}
\end{equation}
\begin{equation}
\sup_{k>2m}\frac{(b_{k}-b_{k-m})^{2}}{k^{-2\alpha}L^{2}(k)}\leq K_{3}
m^{2}\text{ ,} \label{3}
\end{equation}
\begin{equation}
\sum_{j\geq m}j^{-2\alpha}L^{2}(j)l_{j}\leq K_{4}m^{1-2\alpha}L^{2}
(m)l_{m}\text{ .} \label{4}
\end{equation}
and
\begin{equation}
\sum_{j\geq m}j^{-2\alpha}L^{2}(j)\leq K_{4}m^{1-2\alpha}L^{2}(m)\text{ .}
\label{5}
\end{equation}

This is possible by Lemma \ref{coeff} \ and Lemma \ref{Karamata}.

\begin{lemma}
There is a constant $K$ and an integers $N_{0}$ such that for any two integers
$p$ and $q$ with $1\leq p<q\leq n$ with $q-p\geq N_{0}$ and any $n\geq
N_{0}\ $
\begin{equation}
\frac{1}{B_{n}^{2}}var(\sum_{i=p+1}^{q}X_{ni}^{\prime})\leq K(\frac{q}
{n}-\frac{p}{n})^{2-\alpha}\text{ .} \label{tight}
\end{equation}

\end{lemma}

\textbf{Proof.} We shall use $N_{0}$ that was already constructed above. We
start from the decomposition
\begin{gather*}
\sum_{i=p+1}^{q}X_{ni}^{\prime}=\sum_{i=p}^{q-1}b_{q-i}\varepsilon
_{i}I(|\varepsilon_{i}|\leq\eta_{n-i})+\sum_{i=2p-q}^{p-1}(b_{q-i}
-b_{p-i})\varepsilon_{i}I(|\varepsilon_{i}|\leq\eta_{n-i})\\
+\sum_{i\geq q-2p+1}(b_{q+i}-b_{p+i})\varepsilon_{-i}I(|\varepsilon_{-i}
|\leq\eta_{n+i})=I+II+III\text{ .}
\end{gather*}
We shall estimate the variance of each term separately.

Using the fact that $l_{n}$ is increasing and (\ref{1}) we obtain
\begin{align*}
var(I)  &  \leq\sum_{i=p}^{q-1}b_{q-i}^{2}l_{n-i}=\sum_{j=1}^{q-p}b_{j}
^{2}l_{n-q+j}\leq l_{n}(q-p)\max_{1\leq j\leq q-p}b_{j}^{2}\\
&  \leq K_{1}(q-p)^{3-2\alpha}L^{2}(q-p)l_{n}\text{ }.
\end{align*}
Then, by taking into account that $l_{n}$ is increasing, (\ref{1}) and
(\ref{2}) we have
\begin{gather*}
var(II)\leq\sum_{i=2p-q}^{p-1}(b_{q-i}-b_{p-i})^{2}l_{n-i}\leq l_{2n}
2(q-p)\max_{1\leq j\leq2(q-p)}b_{i}^{2}\\
\leq K_{1}K_{2}(q-p)^{3-2\alpha}L^{2}(q-p)l_{n}\text{ .}
\end{gather*}
To estimate the variance of the last term, we use first (\ref{3}) to obtain
\begin{gather*}
var(III)=\sum_{i\geq q-2p+1}(b_{i+q}-b_{i+p})^{2}l_{n+i}\leq\sum
_{j\geq2(q-p)+1}(b_{j}-b_{j-(q-p)})^{2}l_{n+j-q}\\
\leq K_{3}(q-p)^{2}\sum_{j\geq2(q-p)+1}j^{-2\alpha}L^{2}(j)l_{n+j-q}\text{ }.
\end{gather*}
Now, by the monotonicity of $l_{n}$, because $l_{n+j-q}\leq l_{2n}$ for $j\leq
n$ and $l_{n+j-q}\leq l_{2j}$ for $j>n$ by (\ref{2}), (\ref{4}) and (\ref{5})
\begin{align*}
\sum_{j\geq2(q-p)+1}j^{-2\alpha}L^{2}(j)l_{n+j-q}  &  \leq K_{2}
K_{5}(q-p)^{-2\alpha+1}L^{2}(q-p)l_{n}\\
&  +K_{2}K_{4}(q-p)^{-2\alpha+1}L^{2}(q-p)l_{q-p}
\end{align*}
So, for $K_{6}=K_{2}K_{3}(K_{4}+K_{5})$
\[
var(III)\leq K_{6}(q-p)^{3-2\alpha}L^{2}(q-p)l_{n}\text{ .}
\]
Overall we have so far for a certain constant $K_{7}$ that does not depend on
$p$ or $q$,
\begin{equation}
var(\sum_{i=p+1}^{q}X_{ni}^{\prime})\leq K_{7}(q-p)^{3-2\alpha}L^{2}
(q-p)l_{n}\text{ .} \label{Ineqvar}
\end{equation}
By simple algebra, because $1\leq p<q\leq n$ we derive
\[
var(\sum_{i=p+1}^{q}X_{ni}^{\prime})\leq K_{7}(q-p)^{2-\alpha}l_{n}
n^{1-\alpha}L^{2}(n)\max_{1\leq k\leq n}\frac{k^{1-\alpha}}{n^{1-\alpha}}
\frac{L^{2}(k)}{L^{2}(n)}\text{ }.
\]
Finally, by the point 5 of Lemma \ref{Karamata},
\[
var(\sum_{i=p+1}^{q}X_{ni}^{\prime})\leq K_{8}(\frac{q}{n}-\frac{p}
{n})^{2-\alpha}l_{n}n^{3-2\alpha}L^{2}(n)\text{ .}
\]
Therefore, (\ref{tight}) is established by taking into account (\ref{defBn}).
$\lozenge$

\begin{lemma}
Condition (\ref{condtight1}) is satisfied, namely:
\[
\lim_{n\rightarrow\infty}P(\max_{1\leq k\leq n}|X_{nk}^{\prime}-EX_{nk}
^{\prime}|\geq\varepsilon B_{n})=0\text{ .}
\]

\end{lemma}

\textbf{Proof.} We start from

\[
P(\max_{1<k\leq n}|X_{nk}^{\prime}-EX_{nk}^{\prime}|\geq\varepsilon B_{n}
)\leq\frac{1}{\varepsilon^{4}B_{n}^{4}}\sum_{k=1}^{n}E|X_{nk}^{\prime}
-EX_{nk}^{\prime}|^{4}\text{ .}
\]
We use now Rosenthal inequality (Theorem 1.5.13 in de la Pe\~{n}a and Gin\'{e}
1999), which can be easily extended to an infinite sum of independent random
variables, by truncating the sum and passing to the limit. So, there is a
constant $C,$ such that
\[
E|X_{nk}^{\prime}-EX_{nk}^{\prime}|^{4}\leq C\sum_{i=1}^{\infty}a_{i}
^{4}E\varepsilon^{4}I(|\varepsilon|\leq\eta_{n-k+i})+C(\sum_{i=1}^{\infty
}a_{i}^{2}l_{n-k+i})^{2}=I_{k}+II_{k}\text{ .}
\]
By the point 4 of Lemma \ref{csorgo} and (\ref{ordereta}) it follows that
\begin{align*}
a_{i}^{4}E\varepsilon^{4}I(|\varepsilon|  &  \leq\eta_{n-k+i})\ll a_{i}
^{4}(\eta_{n-k+i}^{2})l_{n-k+i}\\
&  \ll i^{-4\alpha}L^{4}(i)(n-k+i)l_{n-k+i}^{2}\text{ .}
\end{align*}
So
\[
\sum_{k=1}^{n}I_{k}\leq\sum_{i=1}^{\infty}i^{-4\alpha}L^{4}(i)\sum_{k=i}
^{n+i}kl_{k}^{2}\ll n^{2}l_{n}^{2}\text{ .}
\]
Then, by simple computations involving the partition of sum in two parts, one
up to $2n$ and the rest, and then using the properties of regularly functions
and the fact that $2\alpha>1$ we obtain
\[
\sum_{k=1}^{n}II_{k}\leq n(\sum_{i=1}^{\infty}i^{-2\alpha}L^{2}(i)l_{n+i}
)^{2}\leq nl_{n}^{2}\text{ .}
\]
Finally by (\ref{defBn}) we notice that
\[
\frac{n^{2}l_{n}^{2}}{B_{n}^{4}}\rightarrow0\text{ .}
\]
$\lozenge$

\textbf{Step 6. Convergence of finite dimensional distributions.}

Let $0\leq t_{1}<t_{2}<\dots<t_{m}\leq1$. We shall show next that the vector
$(W_{n}^{\prime}(t_{j});1\leq j\leq m)$ converges in distribution to the
finite dimensional distributions of a fractional Brownian motion with Hurst
index $3/2-2\alpha$, i.e. of a Gaussian process with covariance structure
$\frac{1}{2}(t^{3-2\alpha}+s^{3-2\alpha}-(t-s)^{3-2\alpha})$ for $s<t$.

By the Cram\'{e}r-Wold device and taking into account (\ref{negl4}) we have to
study the limiting distribution of $\sum_{j=2}^{m}\lambda_{j}(W_{n}^{\prime
}(t_{j})-EW_{n}^{\prime}(t_{j-1}))$, which we express as a weighted sum of
independent random variables. By elementary computations involving similar
arguments used in the proof of step 3, and taking into account (\ref{negl2})
and (\ref{Leap}), we notice that Lyapunov's condition is satisfied and then,
the limiting distribution is normal with the covariance structure that will be
specified next. We compute now the covariance of $W_{n}^{\prime}(s)$ and
$W_{n}^{\prime}(t)$ for $s\leq t$. By simple algebra
\[
cov(W_{n}^{\prime}(t),W_{n}^{\prime}(s))=\frac{1}{2}(var(W_{n}^{\prime
}(t))+var(W_{n}^{\prime}(s))-var(W_{n}^{\prime}(t)-W_{n}^{\prime}(s)))\text{
.}
\]
We analyze now the variance of $W_{n}^{\prime}(t).$ For each $t$ fixed,
$0\leq$ $t\leq1$
\begin{gather*}
var(W_{n}^{\prime}(t))=\frac{1}{B_{n}^{2}}\sum_{i=0}^{[nt]-1}b_{[nt]-i}
^{2}(E\varepsilon_{0}^{2}I(|\varepsilon_{0}|\leq\eta_{n-i})-E^{2}
\varepsilon_{0}I(|\varepsilon_{0}|\leq\eta_{n-i}))\\
+\frac{1}{B_{n}^{2}}\sum_{i\geq1}(b_{[nt]+i}-b_{i})^{2}(E\varepsilon_{0}
^{2}I(|\varepsilon_{0}|\leq\eta_{n+i})-E^{2}\varepsilon_{0}I(|\varepsilon
_{0}|\leq\eta_{n+i}))\text{ .}
\end{gather*}
Taking into account $E\varepsilon_{0}I(|\varepsilon_{0}|\leq\eta
_{n-i})=-E\varepsilon_{0}I(|\varepsilon_{0}|>\eta_{n-i}),$ by Lemma
\ref{csorgo} and Lemma \ref{coeff}, after some computations, we obtain
\[
var(W_{n}^{\prime}(t))\sim\frac{1}{B_{n}^{2}}\sum_{i=0}^{[nt]-1}b_{[nt]-i}
^{2}l_{n-i}+\frac{1}{B_{n}^{2}}\sum_{i\geq1}(b_{[nt]+i}-b_{i})^{2}
l_{n+i}\text{ .}
\]
With a similar proof as of relation (\ref{sumb}) of Lemma \ref{coeff}, for
every $0\leq t\leq1$
\[
var(W_{n}^{\prime}(t))\rightarrow t^{3-2\alpha}
\]
and for every $0\leq s<t\leq1$
\begin{equation}
var(W_{n}^{\prime}(t)-W_{n}^{\prime}(s))\rightarrow(t-s)^{3-2\alpha}\text{
.}\label{st}
\end{equation}
Then
\[
cov(W_{n}^{\prime}(t),W_{n}^{\prime}(s))\rightarrow\frac{1}{2}(t^{3-2\alpha
}+s^{3-2\alpha}-(t-s)^{3-2\alpha})\,\text{,}
\]
that is the desired covariance structure. $\lozenge$

\subsection{Proof of Theorem \textbf{ \ref{selfCLT}}}

We notice that it is enough to prove only the convergence in (\ref{LLN}). Then
(\ref{dcoeff}), (\ref{defBn}) and (\ref{LLN}) imply
\[
B_{n}^{2}\sim c_{\alpha}n^{2}a_{n}^{2}(\sum_{j=1}^{n}X_{j}^{2})/A^{2}\text{
},
\]
which we combine with Theorem \ref{clt}, via Slutsky's theorem, to obtain the
self-normalized part of the theorem. The proof of (\ref{LLN}) will be
decomposed in several steps.

\textbf{Step 1. Truncation.}

Denote $D_{n}^{2}=A^{2}nl_{n}$. Recall the definition (\ref{defprime}) and set
$X_{nj}^{^{\prime\prime}}=X_{j}-X_{nj}^{^{\prime}}$. In order to prove
(\ref{LLN}) it is enough to establish
\begin{equation}
F_{n}=\sum_{j=1}^{n}(X_{nj}^{^{\prime\prime}})^{2}/D_{n}^{2}\overset
{P}{\rightarrow}0 \label{NEG}
\end{equation}
and
\begin{equation}
G_{n}=\sum_{j=1}^{n}(X_{nj}^{\prime})^{2}/D_{n}^{2}\overset{P}{\rightarrow
}1\text{ .} \label{LLNtr}
\end{equation}
To see this we square the decomposition $X_{j}=X_{nj}^{\prime}+X_{nj}
^{^{\prime\prime}}$; then sum with $j$ from $1$ to $n$, and notice that by the
H\"{o}lder inequality
\[
F_{n}-2(G_{n}F_{n})^{1/2}\leq\frac{1}{D_{n}^{2}}\sum_{j=1}^{n}X_{j}^{2}
-G_{n}\leq F_{n}+2(G_{n}F_{n})^{1/2}\text{ .}
\]
\textbf{Step 2. Proof of (\ref{NEG}). }

We start from
\begin{gather*}
\sum_{k=1}^{n}(X_{nk}^{^{\prime\prime}})^{2}=\sum_{k=1}^{n}\sum_{i=1}^{\infty
}a_{i}^{2}\varepsilon_{k-i}^{2}I(|\varepsilon_{k-i}|>\eta_{n-k+i})+\\
2\sum_{k=1}^{n}\sum_{i<j}a_{i}a_{j}\varepsilon_{k-i}I(|\varepsilon_{k-i}
|>\eta_{n-k+i})\varepsilon_{k-j}I(|\varepsilon_{k-j}|>\eta_{n-k+j})=I+II\text{
.}
\end{gather*}
(here and below $\sum_{i<j}$ denotes double summation). By independence,
monotonicity, and the point 3 of Lemma \ref{csorgo}, we easily deduce that
\begin{gather*}
E|II|\leq2\sum_{k=1}^{n}\sum_{i<j}|a_{i}a_{j}|E|\varepsilon_{k-i}
I(|\varepsilon_{k-i}|>\eta_{n-k+i})\varepsilon_{k-j}I(|\varepsilon_{k-j}
|>\eta_{n-k+j})|\\
\leq2n\sum_{i<j}|a_{i}a_{j}|E|\varepsilon|I(|\varepsilon|>\eta_{i}
)E|\varepsilon|I(|\varepsilon|>\eta_{j})=2n\sum_{i<j}|a_{i}a_{j}|o(\eta
_{i}^{-1}l_{i})o(\eta_{j}^{-1}l_{j})\ \text{.}
\end{gather*}
Then, by (\ref{ordereta}), clearly
\[
E|II|\leq n(\sum_{i\geq1}|a_{i}|i^{-1/2}o(l_{i}^{1/2}))^{2}\text{ .}
\]
Because $\sum_{i\geq1}|a_{i}|i^{-1/2}<\infty,$ and $l_{n\text{ }}$is
increasing, it is easy to see that
\[
E|II|=o(nl_{n})=o(D_{n}^{2})\text{ .}
\]
In order to estimate the contribution of the term $I,$ by changing the order
of summation we express this term\ in the following way
\[
I=\sum_{j=1}^{n}(\sum_{i=1}^{j}a_{i}^{2})\varepsilon_{n-j}^{2}I(|\varepsilon
_{n-j}|>\eta_{j})+\sum_{j=n+1}^{\infty}(\sum_{i=j-n+1}^{j}a_{i}^{2}
)\varepsilon_{n-j}^{2}I(|\varepsilon_{n-j}|>\eta_{j})\text{ .}
\]
We implement now the notation
\begin{equation}
A_{nj}^{2}=A_{j}^{2}=\sum_{i=1}^{j}a_{i}^{2}\text{ when }j\leq n\text{ and
}A_{nj}^{2}=\sum_{i=j-n+1}^{j}a_{i}^{2}\text{ when }j>n\text{ .} \label{defA}
\end{equation}
and then we express $I$ as
\[
I=\sum_{j=1}^{\infty}A_{nj}^{2}\varepsilon_{n-j}^{2}I(|\varepsilon_{n-j}
|>\eta_{i})\text{ .}
\]
Clearly $A_{nj}^{2}$ are uniformly bounded by a constant. In addition, by
relation (\ref{propL}), for $j>2n$, these coefficients have the following
order of magnitude
\begin{align}
A_{nj}^{2}  &  \ll n^{2}(j-n)^{-2\alpha-1}\max_{j-n\leq k\leq j}L^{2}(k)\leq
n^{2}(j-n)^{-2\alpha-1}\max_{j/2\leq k\leq j}L^{2}(k)\label{orderA}\\
&  \ll n^{2}(j-n)^{-2\alpha-1}\min_{j/2\leq k\leq j}L^{2}(k)\leq
n^{2}(j-n)^{-2\alpha-1}L^{2}(j)\text{ .}\nonumber
\end{align}
Now, we use first the Khinchin's inequality (see Lemma 1.4.13 in de la
Pe\~{n}a and Gin\'{e}, 1999) followed by the triangle inequality and Lemma
\ref{csorgo}, and relation (\ref{ordereta}) to obtain
\[
E\sqrt{I}\ll E|\sum_{j=1}^{\infty}A_{nj}\varepsilon_{n-j}I(|\varepsilon
_{n-j}|>\eta_{j})|=\sum_{j=1}^{\infty}A_{nj}o(\eta_{j}^{-1}l_{j})=\sum
_{j=1}^{\infty}A_{nj}j^{-1/2}o(l_{j}^{1/2})\text{ .}
\]
We notice that by (\ref{orderA}), the point 4 of Lemma \ref{Karamata}, and the
fact that $\alpha>1/2,$
\begin{align*}
E\sqrt{I}  &  =o(\sqrt{nl_{n}})+n\sum_{j\geq n}^{\infty}j^{-\alpha-1/2}
L^{2}(j+n)j^{-1/2}o(l_{j}^{1/2})\\
&  =o(\sqrt{nl_{n}})+O(n^{-\alpha}L^{2}(n)l_{j}^{1/2})=o(\sqrt{nl_{n}})\text{
.}
\end{align*}
As a consequence, $\sqrt{I/nl_{n}}$ converges in $L_{1}$ to $0$, and so,
$I/D_{n}^{2}$ is convergent to $0$ in probability. By gathering all these
facts we deduce that (\ref{NEG}) holds and the proof is reduced to show that
(\ref{LLNtr}) holds.

\textbf{Step 3. Proof of (\ref{LLNtr}).}

We express the sum of squares as
\begin{gather*}
\sum_{k=1}^{n}(X_{k}^{\prime})^{2}=\sum_{k=1}^{n}\sum_{i=1}^{\infty}a_{i}
^{2}\varepsilon_{k-i}^{2}I(|\varepsilon_{k-i}|\leq\eta_{n-k+i})\\
+2\sum_{k=1}^{n}\sum_{1\leq i<j}a_{i}a_{j}\varepsilon_{k-i}I(|\varepsilon
_{k-i}|\leq\eta_{n-k+i})\varepsilon_{k-j}I(|\varepsilon_{k-j}|\leq\eta
_{n-k+j})\text{ .}
\end{gather*}
We shall show that
\begin{equation}
\frac{1}{D_{n}^{2}}\sum_{k=1}^{n}\sum_{i=1}^{\infty}a_{i}^{2}\varepsilon
_{k-i}^{2}I(|\varepsilon_{k-i}|\leq\eta_{n-k+i})\overset{P}{\rightarrow
}1\text{ } \label{convP}
\end{equation}
and
\begin{equation}
\frac{1}{D_{n}^{2}}\sum_{k=1}^{n}\sum_{1\leq i<j}a_{i}a_{j}\varepsilon
_{k-i}I(|\varepsilon_{k-i}|\leq\eta_{n-k+i})\varepsilon_{k-j}I(|\varepsilon
_{k-j}|\leq\eta_{n-k+j})\overset{P}{\rightarrow}0\text{ .} \label{negl1}
\end{equation}
We establish first (\ref{convP}).

By using the notation (\ref{defA}), we have
\[
\sum_{k=1}^{n}\sum_{i=1}^{\infty}a_{i}^{2}\varepsilon_{k-i}^{2}I(|\varepsilon
_{k-i}|\leq\eta_{n-k+i})=\sum_{i=1}^{\infty}A_{ni}^{2}\varepsilon_{n-i}
^{2}I(|\varepsilon_{n-i}|\leq\eta_{i})\text{ .}
\]
By independence, part 4 of Lemma \ref{csorgo}, relations (\ref{orderA}) and
(\ref{ordereta}), and taking into account that $\alpha>1/2$ we get
\begin{align*}
&  Var(\sum_{i=1}^{\infty}A_{ni}^{2}\varepsilon_{n-i}^{2}I(|\varepsilon
_{n-i}|\leq\eta_{i}))\leq\sum_{i=1}^{\infty}A_{ni}^{4}E\varepsilon
^{4}\mathbf{1}(|\varepsilon|\leq\eta_{i})\\
&  =\sum_{i=1}^{2n}\eta_{i}^{2}o(l_{i})+\sum_{i\geq2n}(i-n)^{-4\alpha-2}
L^{4}(i)n^{4}\eta_{i}^{2}o(l_{i})=o(n^{2}l_{n}^{2})=o(D_{n}^{4})\text{ .}
\end{align*}
So (\ref{convP}) is reduced to showing that
\[
\frac{1}{D_{n}^{2}}\sum_{i=1}^{\infty}A_{ni}^{2}E(\varepsilon^{2}
I(|\varepsilon|\leq\eta_{i})=\frac{1}{D_{n}^{2}}\sum_{i=1}^{\infty}A_{ni}
^{2}l_{i}\rightarrow1\text{ as }n\rightarrow\infty\text{ .}
\]
We divide the sum in three parts, one from $1$ to $n$, one from $n+1$ to $2n$
and the rest of the series. We easily see that by (\ref{orderA}),

\[
\sum_{j=2n+1}^{\infty}A_{nj}^{2}l_{j}\ll\sum_{j=2n+1}^{\infty}(j-n)^{-2\alpha
-1}n^{2}L^{2}(j)l_{j}=o(D_{n}^{2})\text{ .}
\]
Then,
\begin{gather*}
\sum_{j=n+1}^{2n}A_{nj}^{2}l_{j}\ll\sum_{j=n+1}^{2n}(j-n)^{1-2\alpha}l_{j}
\max_{1\leq i\leq n}L(i)\\
\ll(\sum_{k=1}^{n}k^{1-2\alpha}l_{n+k})\max_{1\leq i\leq n}L(i)=o(D_{n}
^{2})\text{ .}
\end{gather*}
Now, by the proof of relation (\ref{rel1}) in Appendix with the only
difference that we replace $a_{i}$ by $a_{i}^{2}$ and so $b_{n}^{2}$ by
$A_{j}^{2}$ we obtain
\begin{equation}
\sum_{j=1}^{n}A_{j}^{2}l_{j}\sim l_{n}\sum_{j=1}^{n}A_{j}^{2}\text{ .}
\label{relA}
\end{equation}
Finally, by (\ref{relA}), the definition of $D_{n}^{2}$ and by the Toeplitz
lemma, (\ref{toeplitz}) in Appendix, it follows
\[
\lim_{n\rightarrow\infty}\frac{1}{D_{n}^{2}}\sum_{j=1}^{n}A_{j}^{2}l_{j}
=\lim_{n\rightarrow\infty}\frac{l_{n}\sum_{j=1}^{n}A_{j}^{2}}{l_{n}nA^{2}
}=\lim_{n\rightarrow\infty}\frac{A_{n}^{2}}{A^{2}}=1\text{ .}
\]
This completes the proof of (\ref{convP}).

We move now to prove (\ref{negl1}). Let $N$ be a fixed positive integer. For
each $1\leq k\leq n$ we divide the sum in two parts:
\begin{align*}
\sum_{j=1}^{N}\sum_{i=1}^{j-1}a_{i}a_{j}\varepsilon_{k-i}I(|\varepsilon
_{k-i}|  &  \leq\eta_{n-k+i})\varepsilon_{k-j}I(|\varepsilon_{k-j}|\leq
\eta_{n-k+j})\\
+\sum_{j>N}\sum_{i=1}^{j-1}a_{i}a_{j}\varepsilon_{k-i}I(|\varepsilon_{k-i}|
&  \leq\eta_{n-k+i})\varepsilon_{k-j}I(|\varepsilon_{k-j}|\leq\eta
_{n-k+j})=I_{k}+II_{k}\text{ .}
\end{align*}
We estimate the variance of the sum of each term separately.

For estimating $var(\sum_{k=1}^{n}II_{k})$ we apply the H\"{o}lder inequality:
\[
var(\sum_{k=1}^{n}II_{k})\leq n\sum_{k=1}^{n}var(\sum_{j>N}\sum_{i=1}
^{j-1}a_{i}a_{j}\varepsilon_{k-i}I(|\varepsilon_{k-i}|\leq\eta_{n-k+i}
)\varepsilon_{k-j}I(|\varepsilon_{k-j}|\leq\eta_{n-k+j}))\text{ .}
\]
By independence, a term corresponding to the combination of indexes
$(k-i_{1},k-j_{1},k-i_{2},k-j_{2})$ with $i_{1}<j_{1}$ has a non-null
contribution if and only if $i_{1}=i_{2}$ and $j_{1}=j_{2},$ leading to
\[
var(\sum_{k=1}^{n}II_{k})\leq n^{2}\sum_{j>N}\sum_{i=1}^{j-1}a_{i}^{2}
a_{j}^{2}l_{n+i}l_{n+j}=(n^{2}l_{n}^{2})o_{N}(1)\text{ ,}
\]
where we used first the monotonicity of $l_{n}$ and in the last part we used
the fact that (by monotonicity, the definition of slowly varying functions and
our notations) $l_{i}\leq l_{2n}\ll l_{n},$ for $i\leq2n$ and $l_{i+n}\leq
l_{3i/2}\ll l_{i}$ for $i>2n$ along with the convergence of the series
$\sum_{i}a_{i}^{2}l_{i}$.

In order to treat the other term we start from
\[
var(\sum_{k=1}^{n}I_{k})=var(\sum_{j=1}^{N}\sum_{i=1}^{j-1}a_{i}a_{j}
\sum_{k=1}^{n}\varepsilon_{k-i}I(|\varepsilon_{k-i}|\leq\eta_{n-k+i}
)\varepsilon_{k-j}I(|\varepsilon_{k-j}|\leq\eta_{n-k+j}))
\]
and then, because we compute the variance of at most $N^{2}$ sums and because
the coefficients $a_{i}$ are bounded, clearly,
\[
var(\sum_{k=1}^{n}I_{k})\ll N^{4}\max_{1\leq i<j\leq N}var(\sum_{k=1}
^{n}\varepsilon_{k-i}I(|\varepsilon_{k-i}|\leq\eta_{n-k+i})\varepsilon
_{k-j}I(|\varepsilon_{k-j}|\leq\eta_{n-k+j}))\text{ .}
\]
We notice now that
\begin{gather*}
var(\sum_{k=1}^{n}\varepsilon_{k-i}I(|\varepsilon_{k-i}|\leq\eta
_{n-k+i})\varepsilon_{k-j}I(|\varepsilon_{k-j}|\leq\eta_{n-k+j}))\\
\leq\sum_{k=1}^{n}E\varepsilon_{k-i}^{2}I(|\varepsilon_{k-i}|\leq\eta
_{n-k+i})E\varepsilon_{k-j}^{2}I(|\varepsilon_{k-j}|\leq\eta_{n-k+j}))\text{
,}
\end{gather*}
since by independence and the fact that $i\neq j$ all the other terms are
equal to $0$. The result is
\[
var(\sum_{k=1}^{n}I_{k})\ll N^{4}\sum_{k=1}^{n}l_{n,n-k+i}l_{n,n-k+j}\ll
N^{4}(nl_{n}^{2})\text{ .}
\]
Overall
\begin{gather*}
\frac{1}{D_{n}^{4}}var(\sum_{k=1}^{n}\sum_{1\leq i<j}\varepsilon
_{k-i}I(|\varepsilon_{k-i}|\leq\eta_{n-k+i})\varepsilon_{k-j}I(|\varepsilon
_{k-j}|\leq\eta_{n-k+j}))\leq\\
\frac{2}{D_{n}^{4}}var(\sum_{k=1}^{n}I_{k})+\frac{2}{D_{n}^{4}}var(\sum
_{k=1}^{n}II_{k})=o_{N}(1)+O(N^{4}\frac{1}{n})\text{.}
\end{gather*}
We conclude that (\ref{negl1}) holds by letting first $n\rightarrow\infty$
followed by $N\rightarrow\infty$. $\lozenge$

\section{Appendix}

We formulate in the first lemma several properties of the slowly varying
function. Their proofs can be found in Seneta (1976).

\begin{lemma}
\label{Karamata} A slowly varying function $l(x)$ defined on $[A,\infty)$ has
the following properties:\newline

\begin{enumerate}
\item There exists $B\geq A$ such that for all $x\geq B,$ $l(x)$ is
representable in the form $l(x)=g(x)\exp(\int_{B}^{x}\frac{a(y)}{y}dy)$, where
$g(x)\rightarrow c_{0}>0$, and $a(x)\rightarrow0$ as $x\rightarrow\infty$. In
addition $a(x)$ is continuous.

\item For $B<c<C<\infty$, $\lim_{x\rightarrow\infty}\frac{l(tx)}{l(x)}=1$
uniformly in $c\leq t\leq C$.

\item For any $\theta>-1$, $\int_{B}^{x}y^{\theta}l(y)dy\mathbb{\sim}
\frac{x^{\theta+1}l(x)}{\theta+1}$ as $x\rightarrow\infty$.

\item For any $\theta<-1$, $\int_{x}^{\infty}y^{\theta}l(y)dy\mathbb{\sim
}\frac{x^{\theta+1}l(x)}{-\theta-1}$ as $x\rightarrow\infty$.

\item For any $\eta>0$, $\sup_{t\geq x}(t^{\eta}l(t))\mathbb{\sim}x^{\eta
}l(x)$ as $x\rightarrow\infty$. Moreover $\sup_{t\geq x}(t^{\eta}
l(t))=x^{\eta}\bar{l}(x)$ where $\bar{l}(x)$ is slowly varying and $\bar
{l}(x)\mathbb{\sim}l(x).$
\end{enumerate}
\end{lemma}

The following lemma contains some equivalent formulation for variables in the
domains of attraction of normal law (\ref{DA}). It is Lemma 1 in
Cs\"{o}rg\H{o} et al (2003); see also Feller (1966).

\begin{lemma}
\label{csorgo} The following statements are equivalent:
\end{lemma}

\begin{enumerate}
\item $l(x)=EX^{2}I(|X|\leq x)$ is a slowly varying function at $\infty$;

\item $P(|X|>x)=o(x^{-2}l(x))$;

\item $E|X|I(|X|>x)=o(x^{-1}l(x))$;

\item $E|X|^{\alpha}I(|X|\leq x)=o(x^{\alpha-2}l(x))$ for $\alpha>2$.
\end{enumerate}

To clarify the behavior of the sequence of normalizer $B_{n}^{2}$ defined by
(\ref{defBn}) we state the following lemma that follows from relations (3.33)
and (3.44) in Kuelbs (1985).

\begin{lemma}
Assume (\ref{DA}) and define $\eta_{n}$ by (\ref{defeta}). Then, $l_{n}
=l(\eta_{n})$ is a slowly varying function at $\infty$.
\end{lemma}

The next lemma is useful to study the variance of partial sums for truncated
random variables.

\begin{lemma}
\label{coeff}Under conditions of Theorem \ref{clt} and with the notation
(\ref{defcalpha}) and (\ref{defbn}) we have
\end{lemma}

\begin{enumerate}
\item The coefficients have the following order of magnitude: There are
constants $C_{1}$ and $C_{2}$ such that for all $n\geq1,$
\begin{equation}
|b_{ni}|\leq C_{1}i^{1-\alpha}|L(i)|\text{ for }i\leq2n\text{ and }
|b_{ni}|\leq C_{2}n(i-n)^{-\alpha}|L(i)|\text{ for }i>2n\ \text{.}
\label{order}
\end{equation}
\begin{equation}
\sum_{i=1}^{\infty}b_{ni}^{2}\mathbb{\sim}c_{\alpha}n^{3-2\alpha}
L^{2}(n)\text{ .}\label{sumb}
\end{equation}

\item The asymptotic equivalence for the variance:
\begin{equation}
\sum_{i\geq1}b_{ni}^{2}l_{i}\mathbb{\sim}l_{n}\sum_{i\geq1}b_{ni}
^{2}\mathbb{\sim}B_{n}^{2}\text{ ,} \label{VAR}
\end{equation}
where $B_{n}^{2}$ is defined by (\ref{defBn}).

\item For any $p\geq1$ and any function $h(x)$ slowly varying at $\infty,$
\begin{equation}
\sum_{i\geq1}|b_{ni}|^{p}i^{-1+p/2}|h(i)|\ll h(n)n^{p(3/2-\alpha)}
L^{p}(n)\text{ .} \label{boundb}
\end{equation}

\end{enumerate}

\textbf{Proof}. The fact that $|b_{ni}|\leq C_{1}i^{1-\alpha}|L(i)|$ for
$i\leq2n$ follows easily by the properties of slowly varying functions listed
in Lemma \ref{Karamata}.

For $i>2n$, by the properties of strong slowly varying functions, for $n$
sufficiently large:
\[
(i-n)^{-\alpha}L(i-n)+...+i^{-\alpha}L(i)\leq\lbrack(i-n)^{-\alpha
}+...+i^{-\alpha}]\max_{i-n\leq j\leq i}L(j)\text{ .}
\]
Then,
\[
\max_{i-n\leq j\leq i}L(j)\leq\max_{i/2\leq j\leq i}L(j)\ll L(i)
\]
since
\begin{equation}
\frac{\max_{m\leq j\leq2m}L(j)}{\min_{m\leq j<2m}L(j)}\rightarrow1\text{ .}
\label{propL}
\end{equation}
The asymptotic equivalence in (\ref{sumb}) is well known. See for instance
Theorem 2 in Wu and Min (2005).

We turn now to show (\ref{VAR}). Let $M$ be a positive integer. We divide the
sum in $3$ parts, one from $1$ to $n$, one from $n+1$ to $nM,$ and the third
one with all the other terms. The idea of the proof is that for $n$ and $M$
large, the sum from $1$ to $nM$ dominates the sum of the rest of the terms.

We treat each of these three sums separately.

By using the definition of $b_{ni}=a_{1}+...+a_{i}=b_{i}$ for $1\leq i\leq n$
by analogy with the point 3 in Lemma \ref{Karamata} we show that
\begin{equation}
\sum_{i=1}^{n}b_{i}^{2}l_{i}\mathbb{\sim}l_{n}\sum_{i=1}^{n}b_{i}^{2}\text{
.}\label{rel1}
\end{equation}
To see this, by the first part of Lemma (\ref{Karamata}) we have $l_{n}
=g_{n}h_{n}$ where $h_{n}=\exp\left(  \int_{B}^{n}\frac{a(y)}{y}dy\right)  $,
$g_{n}\rightarrow c>0$, $a(x)\rightarrow0$ as $x\rightarrow\infty,$ and $a(x)$
is continuous. It is easy to show that
\begin{equation}
h_{n}-h_{n-1}=o(h_{n}/n)\text{ as }n\rightarrow\infty\text{ }\label{propl}
\end{equation}
and also, by the part 3 of the same lemma, we get $\sum_{i=1}^{n-1}b_{i}
^{2}\ll nb_{n}^{2}$ .

Next, we have just to use the well known Toeplitz lemma:
\begin{equation}
\lim_{n\rightarrow\infty}\frac{c_{n}}{d_{n}}=\lim_{n\rightarrow\infty}
\frac{c_{n}-c_{n-1}}{d_{n}-d_{n-1}}\text{ ,} \label{toeplitz}
\end{equation}
provided $d_{n}\rightarrow\infty$ and the limit in the right hand side exists.
Then, it follows that
\[
\lim_{n\rightarrow\infty}\frac{\sum_{i=1}^{n}b_{i}^{2}l_{i}}{l_{n}\sum
_{i=1}^{n}b_{i}^{2}}=\lim_{n\rightarrow\infty}\frac{\sum_{i=1}^{n}b_{i}
^{2}l_{i}}{ch_{n}\sum_{i=1}^{n}b_{i}^{2}}=\lim_{n\rightarrow\infty}\frac
{b_{n}^{2}h_{n}}{h_{n}\sum_{i=1}^{n}b_{i}^{2}-h_{n-1}\sum_{i=1}^{n-1}b_{i}
^{2}}\ \text{.}
\]
We shall show that the limit in the right hand side is equal to $1$. We start
by writing
\[
h_{n}\sum_{i=1}^{n}b_{i}^{2}-h_{n-1}\sum_{i=1}^{n-1}b_{i}^{2}=(h_{n}
-h_{n-1})\sum_{i=1}^{n-1}b_{i}^{2}+b_{n}^{2}h_{n}\text{ .}
\]
Then, by (\ref{propl})
\[
\lim_{n\rightarrow\infty}\frac{b_{n}^{2}h_{n}}{h_{n}\sum_{i=1}^{n}b_{i}
^{2}-h_{n-1}\sum_{i=1}^{n-1}b_{i}^{2}}=\lim_{n\rightarrow\infty}\frac
{b_{n}^{2}h_{n}}{o(h_{n}/n)\sum_{i=1}^{n}b_{i}^{2}+b_{n}^{2}h_{n}}=1\text{ ,}
\]
and (\ref{rel1}) follows.

To treat the second sum, notice that $l_{n}$ is increasing and then
\begin{equation}
\sum_{i=n+1}^{nM}b_{ni}^{2}l_{i}\mathbb{\sim}l_{n}\sum_{i=n+1}^{nM}b_{ni}^{2}
\label{rel2}
\end{equation}
because
\[
l_{n}\sum_{i=n+1}^{nM}b_{ni}^{2}\leq\sum_{i=n+1}^{nM}b_{ni}^{2}l_{i}\leq
l_{nM}\sum_{i=n+1}^{nM}b_{ni}^{2}\text{ ,}
\]
and $l_{n}$ is a function slowly varying at $\infty$ .

We treat now the last sum. By (\ref{sumb}), and Lemma \ref{Karamata}
\[
\sum_{i=nM+1}^{\infty}b_{ni}^{2}l_{i}\ll n^{2}\sum_{i=nM+1}^{\infty
}(i-n)^{-2\alpha}L^{2}(i)l_{i}\ll n^{2}[n(M-1)]^{1-2\alpha}L^{2}
[nM]l_{nM}\text{ .}
\]
We obtain
\begin{equation}
\sum_{i=nM+1}^{\infty}b_{ni}^{2}l_{i}\ll B_{n}^{2}M^{1-2\alpha}\text{ as
}n\rightarrow\infty\text{ .} \label{rel3}
\end{equation}
We combine now the estimates in (\ref{rel1}) and (\ref{rel2}). For $\delta>0$
fixed and $n$ sufficiently large
\[
(1-\delta)l_{n}\sum_{i\geq1}^{nM}b_{ni}^{2}\leq\sum_{i\geq1}^{nM}b_{ni}
^{2}l_{i}\leq(1+\delta)l_{n}\sum_{i\geq1}^{nM}b_{ni}^{2}\text{ .}
\]
Therefore,
\begin{equation}
(1-\delta)l_{n}(\sum_{i\geq1}^{\infty}b_{ni}^{2}-\sum_{i>nM}b_{ni}^{2}
)\leq\sum_{i\geq1}^{\infty}b_{ni}^{2}l_{i}\leq(1+\delta)l_{n}\sum_{i\geq
1}^{nM}b_{ni}^{2}+\sum_{i>nM}b_{ni}^{2}l_{i}\text{ .} \label{IneqB}
\end{equation}
Then, by (\ref{rel3}), for a positive constant $C_{1}$ we have
\[
\lim\sup_{n\rightarrow\infty}\frac{1}{B_{n}^{2}}\sum_{i>nM}b_{ni}^{2}l_{i}
\leq\frac{C_{1}}{M^{2\alpha-1}}\text{ .}
\]
We also know that for a certain positive constant $C_{2}$,
\[
\lim\sup_{n\rightarrow\infty}\frac{1}{B_{n}^{2}}\sum_{i>nM}b_{ni}^{2}\leq
\frac{C_{2}}{M^{2\alpha-1}}\text{ .}
\]
The result follows by dividing (\ref{IneqB}) by $B_{n}^{2}$ and taking first
$\lim\sup$ and also $\lim\inf$ when $n\rightarrow\infty$ followed by
$M\rightarrow\infty$ , and finally we let $\delta\rightarrow0$.

The proof of (\ref{boundb}) is similar and it is sufficient to divide the sum
in only two parts, one from $1$ to $2n$ and the rest. More exactly by using
(\ref{order}),
\[
\sum_{i=1}^{2n}|b_{ni}|^{p}i^{-1+p/2}|h(i)|\ll\sum_{i=1}^{2n}i^{p(1-\alpha
)}|L(i)|^{p}i^{-1+p/2}|h(i)|\ll h(n)n^{p(3/2-\alpha)}L^{p}(n)
\]
and,
\[
\sum_{i\geq2n}|b_{ni}|^{p}i^{-1+p/2}|h(i)|\ll n^{p}\sum_{i\geq2n}
i^{-1+(1/2-\alpha)p}|L(i)|^{p}|h(i)|\ll h(n)n^{p(3/2-\alpha)}L^{p}(n)\text{ .}
\]
The proof is complete. $\lozenge$

Next lemma is a variant of Theorem 12.3 in Billingsley (1968).

\begin{lemma}
\label{Ltight} Assume that $(X_{nk})_{1\leq k\leq n}$ is a triangular array of
centered random variables with finite second moment. For $0\leq m\leq n$ let
$S_{m}=\sum_{j=1}^{m}X_{nj}$ and for $0\leq t\leq1,$ $W_{n}(t)=S_{[nt]}.$
Assume that for every $\varepsilon>0$
\begin{equation}
P(\max_{1\leq i\leq n}|X_{ni}|>\varepsilon)\rightarrow0 \label{condtight1}
\end{equation}
and there is a positive constant $K,$ and an integer $N_{0}$ such that for any
$1\leq p<q\leq n$ with $q-p>N_{0}$ we have
\begin{equation}
E(S_{nq}-S_{np})^{2}\leq K(\frac{q}{n}-\frac{p}{n})^{\gamma}
\label{condtight2}
\end{equation}
for some $\gamma>1.$ Then $W_{n}(t)$ is tight in $D[0,1]$, endowed with
Skorohod topology.
\end{lemma}

\textbf{Proof. }We shall base our proof on a blocking argument. We divide the
variables in blocks of size $N_{0}.$ Let $k=[n/N_{0}].$ For $1\leq j\leq k$
denote $Y_{nj}=\sum_{i=(j-1)N_{0}+1}^{jN_{0}}X_{ni}$ and $Y_{n,k+1}
=\sum_{i=kN_{0}+1}^{n}X_{ni}.$ Define $V_{n}(t)=\sum_{j=1}^{[kt]}Y_{nk}$ .

Then we notice that it is enough to show that $V_{n}(t)$ is tight in $D[0,1]$
because by the fact that $[nt]-[kt]\leq2N_{0}$ and by (\ref{condtight1})
\[
P(\sup_{t}|W_{n}(t)-V_{n}(t)|>\varepsilon)\leq P(\max_{1\leq i\leq n}
|X_{ni}|>\varepsilon/2N_{0})\rightarrow0\text{ .}
\]
By Theorem 8.3 in Billingsley (1968) formulated for random elements of D (see
page 137 in Billingsley, 1968) we have to show that for every $0\leq t\leq1$
and $\varepsilon>0$ fixed,

\[
\lim_{\delta\searrow0}\lim\sup_{n\rightarrow\infty}\frac{1}{\delta}
P(\max_{[kt]\leq j\leq\lbrack k(t+\delta)]}|\sum_{i=[kt]}^{j}Y_{ni}
|\geq\varepsilon)=0\text{ .}
\]
By Theorem 12.2 in Billingsley (1968), because $\gamma>1,$ there is a constant
$K$ such that
\[
P(\max_{[kt]\leq j\leq\lbrack k(t+\delta)]}|\sum_{i=[kt]}^{j}Y_{ni}
|\geq\varepsilon)\leq\frac{K}{\varepsilon^{2}}([\frac{k(t+\delta)N_{0}}
{n}]-[\frac{ktN_{0}}{n}])^{\gamma}
\]
and the result follows by multiplying with $1/\delta$ and passing to the limit
with $n\rightarrow\infty$ and then with $\delta\rightarrow0$. $\lozenge$

\section{Acknowledgement}

The authors are grateful to the referees for carefully reading the paper and
for numerous suggestions that significantly improved the presentation of the paper.