text:
\begin{document}

\title{A quasimartingale characterization of $p$-stable type Banach spaces.}

\author{Florian HECHNER $^1$ and Bernard HEINKEL$^2$\\
$^1$Corresponding author\\IRMA, University of Strasbourg and CNRS,\\ 7 rue Ren\'e-Descartes, 67084 STRASBOURG CEDEX\\
hechner@math.unistra.fr\\[2pt]
$^2$IRMA, University of Strasbourg and CNRS,\\ 7 rue Ren\'e-Descartes, 67084 STRASBOURG CEDEX\\
heinkel@math.unistra.fr}

\maketitle

\begin{abstract}
We characterize Banach spaces $\brond$ of stable-type $p$ ($1<p<2$) by the property that for every sequence $({\bf X}_i)$ of ${\brond}$-valued random variables, independent, centered and fulfilling some integrability assumption, the sequence $\left(\frac{{\bf X}_1+\cdots+{\bf X}_n}{n^{1/p}}\right)$ is a quasimartingale.

{\bf AMS Subject Classification:} 46B09, 60B12, 60G48.

{\bf Key Words and Phrases:} Marcinkiewicz-Zygmund Law of Large Numbers; Quasimartingales; Type of a Banach space; Stable type of a Banach space
\end{abstract}

\section{Introduction}

~\guillegauche How to characterize the regularity of a Banach space $(\brond,\|\cdot\|)$ by the fact that a kind of classical strong law of large numbers (SLLN) holds for $\brond$-valued random variables (r.v.)?\guilledroit\ is a well know problem. Two kinds of regular Banach spaces -- the spaces of Rademacher type $p$ and the spaces of stable type $p$ -- have been nicely characterized in that way (see chapter 9 in \cite{Ledoux1}).

Here our goal will be to show that the SLLN characterization of spaces of stable type $p$ ($1<p<2$) can be made more precise in supposing that the \guillegauche normalized sums\guilledroit\ obeying the SLLN have even a quasimartingale behaviour.

To begin with, we recall some definitions.

In the sequel, $(\brond,\|\cdot\|)$ will be a real separable Banach space, equipped with its Borel $\sigma$-field $B$. A $\brond$-valued r.v. $\va X$ is a measurable function defined on a probability space $(\Omega,\trond,\PM)$ with values in $(\brond,B)$. Such a r.v. is said to be (strongly) integrable if $\EM\|\va X\|<+\infty$ and $\forall f\in\brond^\prime,\ \EM f(\va X)=0$; this is denoted by $\EM(\va X)=0$.

Let $p\geq 1$ be given. The weak-$\ell_p$ norm of a sequence $a:=(a_1,\ldots,a_n)$ of real numbers is defined as follows~:
$$
\|a\|_{p,\infty}:=\sup\limits_{t>0}{\left(t^p\card(i:|a_i|>t)\right)}^\frac{1}{p}=\sup\limits_{k=1}^na_k^*k^{1/p},
$$
where $(a_1^*,\ldots,a_n^*)$ denotes the non-increasing rearrangement of the sequence $(|a_1|,\ldots,|a_n|)$.

Let now $(\varepsilon_k)$ be a sequence of independent Rademacher random variables $\left(\text{that is}\ \PM(\varepsilon_k=1)=\PM(\varepsilon_k=-1)=\frac{1}{2}\right)$.

Rademacher type $p$ spaces and stable type $p$ spaces are defined as follows~:

\begin{definition}\label{defi1}~
\begin{enumerate}
\item Let $1<p<2$. The Banach space $(\brond,\|\cdot\|)$ is of Rademacher type $p$ if there exists a constant $c(p)>0$ such that for every finite sequence $(x_i)$ in $\brond$~:
$$
{\left(\EM\left\|\sum\limits_{i=1}^n\varepsilon_i x_i\right\|^p\right)}^{\frac{1}{p}}\leq c(p){\left(\sum\limits_{i=1}^n\|x_i\|^p\right)}^{\frac{1}{p}}.
$$
\item Let $1\leq p\leq 2$. The Banach space $(\brond,\|\cdot\|)$ is of stable type $p$ if there exists a constant $C(p)>0$ such that for every finite sequence $(x_i)$ in $\brond$~:
$$
\EM\left\|\sum\limits_{i=1}^n\varepsilon_i x_i\right\|\leq C(p)\left\|\left(\|x_1\|,\ldots,\|x_n\|\right)\right\|_{p,\infty}.
$$
\end{enumerate}
\end{definition}

\begin{remark}\label{rem1}~
\begin{enumerate}
\item If $(\brond,\|\cdot\|)$ is of stable type $p$, then there exists $q>p$, such that $(\brond,\|\cdot\|)$ is also of stable type $q$ \cite{Maureypisier}.
\item If $(\brond,\|\cdot\|)$ is of stable type $p$, it is also of Rademacher type $p$ \cite{Pisier1}.
\item The above definition of stable type is not the classical one (which involves standard stable r.v.), but an equivalent statement (see for instance \cite{Ledoux1}, proposition 9.12) which will be used in our proofs.
\end{enumerate}
\end{remark}

\begin{remark}~\\
Let $(\brond,\|\cdot\|)$ be a Banach space and $1\leq p\leq 2$. If $\brond$ is of (Rademacher) type $p$, there exists a constant $c(p)$ such that for every finite sequence $(\va X_i)$ with $\|\va X_i\|\in L^p$~:
\begin{equation}\label{relation01}
\EM{\left\|\sum\limits_{i=1}^n\va X_i\right\|}^p\leq c(p)\sum\limits_{i=1}^n\EM\|\va X_i\|^p.
\end{equation}
\end{remark}

The stable type $p$ has been characterized in terms of Marcinkiewicz-Zygmund like SLLN~:

\begin{theorem}[Maurey-Pisier \cite{Maureypisier}]\label{theo2}~\\
Let $1<p<2$. The following two properties are equivalent~:
\begin{enumerate}
\item $(\brond,\|\cdot\|)$ is of stable type $p$.
\item For every bounded sequence $(x_i)$ in $\brond$, the sequence $\left(\frac{1}{n^{1/p}}\sum\limits_{k=1}^n\varepsilon_k x_k\right)$ converges a.s. to 0.
\end{enumerate}
\end{theorem}

\begin{theorem}[Woyczynski \cite{Woyczynski}]\label{theo3}~\\
Let $1<p<2$. The following two properties are equivalent~:
\begin{enumerate}
\item $(\brond,\|\cdot\|)$ is of stable type $p$.
\item For every sequence $(\va X_i)$ of independent, strongly centered $\brond$-valued r.v. for which there exists a nonnegative r.v. $\xi$ with $\EM\xi^p<+\infty$ such that~:
$$
\exists c>0,\ \forall t>0,\ \forall i\in\NM^*,\ \PM(\|\va X_i\|>t)\leq c\PM(\xi>t),
$$
the sequence $\left(\frac{\va S_n}{n^{1/p}}\right)$ converges a.s. to 0, where, as usual, $\va S_n:=\va X_1+\cdots+\va X_n$.
\end{enumerate}
\end{theorem}

In this paper, we will prove a result in the same spirit as Woyczynski's result, but in which the SLLN behaviour of the sequence $\left(\frac{\va S_n}{n^{1/p}}\right)$ is even a quasimartingale behaviour.

\section{A quasimartingale characterization of spaces of stable type $p$.}

We start this section by defining the quasimartingale behaviour of $\left(\frac{\va S_n}{n^{1/p}}\right)$~:

\begin{definition}\label{defi4}~\\
Let $(\va X_k)$ be a sequence of independent, strongly centered $(\brond,\|\cdot\|)$-valued r.v. Denote $\va S_n:=\va X_1+\cdots+\va X_n$ and $\frond_n:=\sigma(\va X_1,\ldots,\va X_n)$. Let $p\in]1,2[$. The sequence $\left(\frac{\va S_n}{n^{1/p}},\frond_n\right)$, or simplier $\left(\frac{\va S_n}{n^{1/p}}\right)$, is a quasimartingale if~:
\begin{equation}\label{1etoile}
\sum\limits_{n=1}^{+\infty}\EM\left\|\left.\EM\left(\frac{\va S_{n+1}}{{(n+1)}^{1/p}}-\frac{\va S_n}{n^{1/p}}\right|\frond_n\right)\right\|<+\infty.
\end{equation}
\end{definition}

\begin{remark}\label{rem5}~\\
Since the r.v. $(\va X_k)$ are independent and centered, condition (\ref{1etoile}) is equivalent to~:
$$
\sum\limits_{n=1}^{+\infty}\frac{\EM\|\va S_n\|}{n^{1+1/p}}<+\infty.
$$
\end{remark}

Now we are able to state our characterization of spaces of stable type $p$ $(1<p<2)$~:

\begin{theorem}\label{theo6}~\\
Let $1<p<2$. The following two properties are equivalent~:
\begin{enumerate}
\item $\brond$ is a stable type $p$ space;
\item For every sequence $(\va X_n)$ of independent, strongly centered r.v., such that
\begin{equation}\label{2etoiles}
\int_0^{+\infty}g^{1/p}(t)dt<+\infty,
\end{equation}
where
$$
\forall t>0,\ g(t):=\sup\limits_{n\geqslant 1}\PM(\|\va X_n\|>t),
$$
$\left(\frac{\va S_n}{n^{1/p}}\right)$ is a quasimartingale.
\end{enumerate}
\end{theorem}

\begin{remark}\label{rem2}~
\begin{enumerate}
\item Condition (\ref{2etoiles}) is not surprising~: indeed, in the i.i.d. case, it can be written~:
\begin{equation}\label{Etoile}
\int_0^{+\infty}\PM^{1/p}(\|\va X\|>t)dt<+\infty,
\end{equation}
which condition is necessary for $\left(\frac{\va S_n}{n^{1/p}}\right)$ being a quasimartingale in every Banach space $\brond$ (see \cite{hechnerheinkel}, proposition 3).
\item Property (\ref{Etoile}) implies that $\EM\|\va X\|^p<+\infty$ (see \cite{hechnerheinkel}, remark 4).
\item There exists a small class of r.v. $\va X$ such that $\EM\|\va X\|^p<+\infty$ and for which (\ref{Etoile}) does not hold (see example 1 in \cite{hechnerheinkel}).
\end{enumerate}
\end{remark}

So -- comparing theorems \ref{theo3} and \ref{theo6} -- one sees that the price to pay for getting a quasimartingale behaviour for $\left(\frac{\va S_n}{n^{1/p}}\right)$ instead of a simple a.s. convergence to 0, is to sharpen a little bit the hypothesis $\EM\xi^p<+\infty$ of theorem \ref{theo3}.

\bigskip

\proof~

In the sequel, $c_k$ will denote positive constants which precise value does not matter.

Let us show the implication $1\implique 2$.

First consider the special case where there exists $M>0$ such that~: $\forall t>M,\ g(t)=0$. Then $\forall k,\ \|\va X_k\|\leq M$ a.s. The space $\brond$ being of Rademacher type $q$ for some $q>p$ by comments 1) and 2) following definition \ref{defi1}, and using relation (\ref{relation01}), one has~:
\begin{align*}
\frac{\EM\|\va S_n\|}{n^{1+1/p}}&\leq\frac{c(q)}{n^{1+1/p}}{\left(\sum\limits_{k=1}^n\EM\|\va X_k\|^q\right)}^{\frac{1}{q}}\leq\frac{c(q)M n^\frac{1}{q}}{n^{1+\frac{1}{p}}}
\end{align*}
and the series having general terms $\frac{\EM\|\va S_n\|}{n^{1+\frac{1}{p}}}$ converges.

\medskip

From now we suppose that $\forall t>0, g(t)>0$.

By a classical symmetrization argument it suffices to consider the case of symmetrically distributed r.v. $(\va X_k)$. For showing that under condition (\ref{2etoiles}) the series having general term $\frac{\EM\|\va S_n\|}{n^{1+\frac{1}{p}}}$ converges, one will split each r.v. $\va X_1,\ldots,\va X_n$ involved in the sum $\va S_n$ into two parts $\va U_{n,k}$ and $\va V_{n,k}$ by truncating $\va X_k$ at a suitable level $v_n$.

For defining $v_n$, one first notices that, $g$ being decreasing, one has~:
$$
\sup\limits_{t>0}t^pg(t)\leq{\left(\int_0^{+\infty}g^{\frac{1}{p}}(x)dx\right)}^p<+\infty.
$$

Furthermore, multiplying the $\va X_k$ by a suitable constant if necessary, one can suppose without loss of generality that~:
\begin{equation}\label{3etoiles}
\sup\limits_{t>0}t^pg(t)\leq 1.
\end{equation}

Now define $v_n:=\inf\left(t>0|g(t)\leq\frac{1}{n}\right)$.

It follows from the definition of $g$ and (\ref{3etoiles}) that~:
$$
v_n\leq n^{1/p}\qquad\text{and}\qquad g(v_n)\leq\frac{1}{n}.
$$

For every $n\in\NM^*$ and $k=1,\ldots,n$, one considers the following centered r.v. (by symmetry)~:
$$
\va U_{n,k}:=\va X_k\va 1_{(\|\va X_k\|\leq v_n)} \qquad\text{and}\qquad \va V_{n,k}:=\va X_k\va 1_{(\|\va X_k\|> v_n)},
$$
and the associated sums~:
$$
\va A_n:=\sum\limits_{k=1}^n\frac{\va U_{n,k}}{n^{1+\frac{1}{p}}}\qquad\text{and}\qquad\va B_n:=\sum\limits_{k=1}^n\frac{\va V_{n,k}}{n^{1+\frac{1}{p}}}.
$$

For showing that the series having general terms $\frac{\EM\|\va S_n\|}{n^{1+\frac{1}{p}}}$ converges, one will show that~:
\begin{equation}\label{4etoiles}
\sum\limits_{n=1}^{+\infty}\EM\|\va A_n\|<+\infty
\end{equation}
and
\begin{equation}\label{5etoiles}
\sum\limits_{n=1}^{+\infty}\EM\|\va B_n\|<+\infty.
\end{equation}

We first prove (\ref{4etoiles}).

By remark $1.$ following definition \ref{defi1}, there exists $q>p$ such that $(\brond,\|\cdot\|)$ is also $q$--stable.

Now suppose that the r.v. $\va X_k$ are defined on a probability space $(\Omega,\trond,\PM)$ and consider $(\varepsilon_k)$ a sequence of independent Rademacher r.v. defined on another probability space $(\Omega^\prime,\trond^\prime,\PM^\prime)$. By symmetry, one has~:
$$
\EM\|A_n\|=\int_\Omega\left(\int_{\Omega^\prime}\frac{1}{n^{1+\frac{1}{p}}}\left\|\sum\limits_{k=1}^n\varepsilon_k(\omega^\prime)\va U_{n,k}(\omega)\right\|d\PM^\prime(\omega^\prime)\right)d\PM(\omega).
$$
By application of the definition of the stable type $q$, one obtains~:
$$
\EM\|A_n\|\leq\frac{C(q)}{n^{1+\frac{1}{p}}}\EM\left(\|(\|\va U_{n,1}\|,\ldots,\|\va U_{n,n}\|)\|_{q,\infty}\right).
$$
For bounding the tails of the weak-$\ell_p$ norm of a sequence of positive, independent r.v., we will use the following classical result due to Marcus Pisier \cite{Marcuspisier}~:
\begin{lemma}\label{lemme6}~\\
For positive valued, independent r.v. $\xi_1,\ldots,\xi_n$, one has~:
\begin{equation}\label{relp6}
\forall q\geq 1,\ \forall u>0,\ \PM\left(\|(\xi_1,\ldots\xi_n)\|_{q,\infty}>u\right)\leq\frac{2e}{u^q}\Delta(\xi_1,\ldots,\xi_n),
\end{equation}
where $\Delta(\xi_1,\ldots,\xi_n)=\sup\limits_{t>0}\left(t^q\sum\limits_{k=1}^n\PM(\xi_k>t)\right)$.
\end{lemma}

For simplicity, denote $\Delta_n$ the quantity $\Delta(\|\va U_{n,1}\|,\ldots,\|\va U_{n,n}\|)$, and notice that by application of lemma \ref{lemme6}~:
\begin{align*}
\EM\left(\|(\|\va U_{n,1}\|,\ldots,\|\va U_{n,n}\|)\|_{q,\infty}\right)
&=\int_0^{+\infty}\PM\left(\|(\|\va U_{n,1}\|,\ldots,\|\va U_{n,n}\|)\|_{q,\infty}>u\right)du\\
&\leq\Delta_n^{\frac{1}{q}}+\int_{\Delta_n^{\frac{1}{q}}}^{+\infty}\frac{2e}{u^q}\Delta_ndu\\
&\leq\cpp\con\Delta_n^{\frac{1}{q}}
\end{align*}

So the proof of (\ref{4etoiles}) reduces to the following lemma~:

\begin{lemma}\label{lemma7}
$$
\sum\limits_{n=1}^{+\infty}\frac{\Delta_n^{\frac{1}{q}}}{n^{1+\frac{1}{p}}}<+\infty
$$
\end{lemma}

Proof of lemma \ref{lemma7}~:\\

One first notices that~:
\begin{align*}
\Delta_n&\leq\sup\limits_{t\leq v_n}t^q\sum\limits_{k=1}^n\PM(\|\va X_k\|>t)
\leq\sup\limits_{t\leq v_n}nt^qg(t)
\leq n{\left(\int_0^{v_n}g^{\frac{1}{q}}(u)du\right)}^{q},
\end{align*}
the last inequality following from the fact that $g$ is decreasing.

For concluding the proof of lemma \ref{lemma7} it remains to check that the series with general term  $a_n:=\frac{1}{n^{1+\frac{1}{p}-\frac{1}{q}}}\int_0^{v_n}g^\frac{1}{q}(u)du$ converges.

First observe that~:
$$
\sum\limits_{n=1}^{+\infty}a_n\leq\sum\limits_{n=1}^{+\infty}\frac{1}{n^{1+\frac{1}{p}-\frac{1}{q}}}\sum\limits_{j=0}^n\int_{v_j}^{v_{j+1}}g^{\frac{1}{q}}(u)du,
$$
where $v_0:=0$, and then exchange the summations in $n$ and $j$~:
$$
\sum\limits_{n=1}^{+\infty}a_n\leq\cpp\con\left(v_1+\sum\limits_{j=1}^{+\infty}\frac{1}{j^{\frac{1}{p}-\frac{1}{q}}}\left(\int_{v_j}^{v_{j+1}}g^{\frac{1}{q}}(u)du\right)\right),
$$
so, by the definition of $v_{j+1}$~:
$$
\sum\limits_{n=1}^{+\infty}a_n\leq\cpp\con\left(v_1+\int_0^{+\infty}g^{\frac{1}{p}}(u)du\right),
$$
which concludes the proof of lemma \ref{lemma7}.

Now we are going to prove (\ref{5etoiles}).

First notice the following chain of inequalities~:
\begin{align*}
\EM\|\va B_n\|&\leq\sum\limits_{k=1}^n\frac{\EM\|\va V_{n,k}\|}{n^{1+\frac{1}{p}}}
=\sum\limits_{k=1}^n\int_0^{+\infty}\PM\left(\frac{\|\va V_{n,k}\|}{n^{1+\frac{1}{p}}}>u\right)du\\
&\leq\sum\limits_{k=1}^n\frac{v_n}{n^{1+\frac{1}{p}}}\PM(\|\va X_k\|>v_n)+\frac{n}{n^{1+\frac{1}{p}}}\int_{v_n}^{+\infty}g(u)du\\
&\leq\frac{nv_n}{n^{1+\frac{1}{p}}}g(v_n)+\frac{1}{n^{\frac{1}{p}}}\int_{v_n}^{+\infty}g(u)du
\leq\frac{v_n}{n^{1+\frac{1}{p}}}+\frac{1}{n^{\frac{1}{p}}}\int_{v_n}^{+\infty}g(u)du.
\end{align*}

Therefore, for proving (\ref{5etoiles}), it suffices to check that condition (\ref{2etoiles}) implies the convergence of the two series with general terms $\frac{v_n}{n^{1+\frac{1}{p}}}$ and $\frac{1}{n^{\frac{1}{p}}}\int_{v_n}^{+\infty}g(u)du$.

\begin{lemma}\label{lemme8}~\\
If (\ref{2etoiles}) is fulfilled, then $\sum\limits_{n=1}^{+\infty}\frac{v_n}{n^{1+\frac{1}{p}}}<+\infty$.
\end{lemma}

Proof of lemma \ref{lemme8}~:

For $j\in\NM^*$, one denotes $t_j:=\frac{v_j+v_{j+1}}{2}$. Then~:
\begin{align}\label{6etoiles}
\int_0^{+\infty}g^\frac{1}{p}(u)du&\geq\sum\limits_{j=1}^{+\infty}\int_{v_j}^{t_j}g^\frac{1}{p}(t)dt
\geq\sum\limits_{j=1}^{+\infty}\frac{1}{{(j+1)}^{\frac{1}{p}}}\frac{v_{j+1}-v_j}{2}.
\end{align}
Now observe that~:
\begin{align}\label{7etoiles}
\sum\limits_{j=1}^{n-1}\frac{1}{{(j+1)}^{\frac{1}{p}}}(v_{j+1}-v_j)&=-\frac{v_1}{2^{\frac{1}{p}}}+\sum\limits_{j=2}^{n-1}v_j\left(\frac{1}{j^{\frac{1}{p}}}-\frac{1}{{(j+1)}^\frac{1}{p}}\right)+\frac{v_n}{n^\frac{1}{p}}
\end{align}
As~:
$$
\frac{v_n}{2}g^\frac{1}{p}(v_n)\leq\int_{\frac{v_n}{2}}^{v_n}g^{\frac{1}{p}}(u)du,
$$
one gets $\lim\limits_{n\to+\infty} v_ng^{\frac{1}{p}}(v_n)=0$ and also $\lim\limits_{n\to+\infty}\frac{v_n}{n^{\frac{1}{p}}}=0$.

As $\frac{1}{j^\frac{1}{p}}-\frac{1}{{(j+1)}^{\frac{1}{p}}}\geq\frac{\cpp\con}{j^{1+\frac{1}{p}}}$, it follows from (\ref{6etoiles}) and (\ref{7etoiles}) that the series having general term $\frac{v_n}{n^{1+\frac{1}{p}}}$ converges.

For completing the proof of the implication $1\implique 2$ of theorem \ref{theo6}, it remains to check~:

\begin{lemma}\label{lemme9}~\\
Under (\ref{2etoiles}), one has $\sum\limits_{n=1}^{+\infty}\frac{1}{n^{\frac{1}{p}}}\int_{v_n}^{+\infty}g(u)du<+\infty$.
\end{lemma}

Proof of lemma \ref{lemme9}.

Let us write~:

$$
\alpha:=\sum\limits_{n=1}^{+\infty}\frac{1}{n^\frac{1}{p}}\int_{v_n}^{+\infty}g(u)du=\sum\limits_{n=1}^{+\infty}\frac{1}{n^\frac{1}{p}}\sum\limits_{j=n}^{+\infty}\int_{v_j}^{v_{j+1}}g(u)du.
$$
By exchanging the summations in $n$ and $j$, one gets~:
\begin{align*}
\alpha&=\sum\limits_{j=1}^{+\infty}\left(\int_{v_j}^{v_{j+1}}g(u)du\right)\sum\limits_{n=1}^j\frac{1}{n^\frac{1}{p}}
\leq\cpp\con\sum\limits_{j=1}^{+\infty}\left(\int_{v_j}^{v_{j+1}}g^\frac{1}{p}(u)du\right)\frac{j^{1-\frac{1}{p}}}{j^{1-\frac{1}{p}}}\\
&\leq\cpp\con\int_0^{+\infty}g^\frac{1}{p}(u)du.
\end{align*}

Let us now show the converse implication $2\implique 1$ of theorem \ref{theo6}.

We first show a general property which is of independent interest~:

\begin{proposition}\label{prop10}~\\
Let $(\va Y_n)$ be a sequence of independent strongly centered r.v. with values in a general Banach space $(\brond,\|\cdot\|)$. Denote by $\va T_n$ the sum $\va Y_1+\ldots+\va Y_n$ and by $\grond_n$ the $\sigma$-field $\sigma(\va Y_1,\ldots,\va Y_n)$. If $\left(\frac{\va T_n}{n^{\frac{1}{p}}},\grond_n\right)$ is a quasimartingale, then $\left(\frac{\va T_n}{n^{\frac{1}{p}}}\right)$ converges a.s. to 0.
\end{proposition}

Proof of proposition \ref{prop10}~:

As noticed earlier, if $\left(\frac{\va T_n}{n^{\frac{1}{p}}}\right)$ is a quasimartingale, then~:
\begin{equation}\label{rel5}
\sum\limits_{n=1}^{+\infty}\frac{\EM\|\va T_n\|}{n^{1+\frac{1}{p}}}<+\infty.
\end{equation}
By Jensen's inequality~:
$$
\forall N\in\NM^*,\ \sum\limits_{n=N}^{+\infty}\frac{\EM\|\va T_n\|}{n^{1+\frac{1}{p}}}\geq\EM\|\va T_N\|\sum\limits_{n=N}^{+\infty}\frac{1}{n^{1+\frac{1}{p}}}\geq\cpp\con\frac{\EM\|\va T_N\|}{N^\frac{1}{p}},
$$
so by (\ref{rel5}),
\begin{equation}\label{rel6}
\lim\limits_{n\to+\infty}\frac{\EM\|\va T_n\|}{n^{\frac{1}{p}}}=0.
\end{equation}
By the conditionnal version of Jensen's inequality~:
\begin{equation}\label{rel7}
\forall n\in\NM^*,\ \EM\left(\|\va T_{n+1}\||\grond_n\right)\geq\|\va T_n\|,
\end{equation}
so~:
\begin{align*}
\sum\limits_{n=1}^N\EM\left|\EM\left(\left.\frac{\|\va T_{n+1}\|}{{(n+1)}^{\frac{1}{p}}}-\frac{\|\va T_n\|}{n^\frac{1}{p}}\right|\grond_n\right)\right|&\leq\sum\limits_{n=1}^N\EM\left|\EM\left(\left.\frac{\|\va T_{n+1}\|-\|\va T_n\|}{{(n+1)}^{\frac{1}{p}}}\right|\grond_n\right)\right|\\+&\cpp\con\sum\limits_{n=1}^N\frac{\EM\|\va T_n\|}{n^{1+\frac{1}{p}}}
\end{align*}
and by (\ref{rel7})~:
\begin{align*}
\sum\limits_{n=1}^N\EM\left|\EM\left(\left.\frac{\|\va T_{n+1}\|}{{(n+1)}^{\frac{1}{p}}}-\frac{\|\va T_n\|}{n^\frac{1}{p}}\right|\grond_n\right)\right|
&\leq\EM\left(\frac{\|\va T_{N+1}\|}{{(N+1)}^{\frac{1}{p}}}\right)+\cpp\con\sum\limits_{n=1}^N\frac{\EM\|\va T_n\|}{n^{1+\frac{1}{p}}}.
\end{align*}
Finaly, by (\ref{rel5}) and (\ref{rel6}), the sequence $\left(\frac{\|\va T_n\|}{n^{1+\frac{1}{p}}}\right)$ is a positive quasimartingale. Therefore, thanks to theorem 9.4 in \cite{Metivier1}, it converges a.s. to a limit, which, by (\ref{rel6}) is necessary 0.

This concludes the proof of proposition \ref{prop10}.

Let us now come back to the proof of the implication $(2)\implique (1)$ of theorem \ref{theo6}.

Let $(\varepsilon_k)$ be a sequence of independent Rademacher r.v. and $(x_k)$ be a bounded sequence of elements in $\brond$. Defining $M:=\sup\|x_k\|$, $\va X_k:=\varepsilon_k x_k$, one gets~:
$$
\forall t>M,\ g(t)=\sup\limits_{k}\PM(\|\va X_k\|>t)=0,
$$
so condition (\ref{2etoiles}) holds. Therefore $\frac{1}{n^{\frac{1}{p}}}\sum\limits_{k=1}^n\varepsilon_k x_k$ is a quasimartingale, which by proposition \ref{prop10} converges a.s. to 0. The $p$-stability of the space $(\brond,\|\cdot\|)$ then follows from theorem \ref{theo2}.

\section{What happens when $p=1$?}

It is natural to wonder if the spaces of stable type 1 (see \cite{hechner3} for the definition of stable type 1) also admit a \guillegauche quasimartingale characterization \guilledroit. In fact it is the case, by theorem 6 in \cite{hechner3}, which can be reformulated as follows~:

\begin{theorem}
Let $\brond$ be a Banach space. The following two properties are equivalent~:
\begin{enumerate}
\item $\brond$ is of stable type 1.
\item For every sequence $({\bf X}_n)$ of independent, strongly centered r.v., such that
\begin{equation}
\int_0^{+\infty}g(t)dt<+\infty,
\end{equation}
where
$$\forall t>0,\ g(t):=\sup\limits_{n\geqslant 1}\PM(\|{\bf X}_n\|\ln(1+\|{\bf X}_n\|)>t),$$
$\left(\frac{{\bf S}_n}{n}\right)$ is a quasimartingale.
\end{enumerate}
\end{theorem}

config: arxiv


text:
\section{Conclusions and future work}\label{sec:conclusions}

We have shown that explicit MPC formulations can be exactly represented by deep neural networks with rectifier units as activation functions and included explicit bounds on the dimensions of the required neural networks. The choice of deep networks is especially interesting for the representation of explicit MPC laws as the number of regions that deep networks can represent grows exponentially with their depth.
This notion was exploited to propose an approximation method for explicit MPC solutions.

Stochastic verification techniques have been used to ensure constraint satisfaction if the neural network is used to approximate, and not to exactly represent, the explicit MPC law within a safe set.
Simulation results show that the proposed deep learning-based explicit MPC achieves better performance than other approximate explicit MPC methods with significantly smaller memory requirements.
This significant reduction of the memory footprint enabled the deployment of the proposed deep neural network controllers on a low-power embedded device with constrained resources.

Future work includes the design of stability guaranteeing formulations and the computation of safe sets which allow to predict the probability and the magnitude of constraint violations via gaussian process regression.