text:
\begin{document}

\begin{center}
{\large{\sc Stochastic approximations to the Pitman--Yor process}}
\bigskip

Julyan Arbel, Pierpaolo De Blasi, Igor Pr\"{u}nster
\bigskip

\end{center}

\begin{abstract}
In this paper we consider approximations to the popular Pitman--Yor process obtained by truncating the stick-breaking representation. The truncation is determined by a random stopping rule

that achieves an almost sure control on the approximation error in total variation distance. We derive the asymptotic distribution of the random truncation point as the approximation error $\epsilon$ goes to zero in terms of a polynomially tilted positive stable distribution. The practical usefulness and effectiveness of this theoretical result is demonstrated by devising a sampling algorithm to approximate functionals of the $\epsilon$-version of the Pitman--Yor process.

\end{abstract}

\textbf{Keywords}:
stochastic approximation;
asymptotic distribution;
Bayesian Nonparametrics;
Pitman--Yor process;
random functionals;
random probability measure;
stopping rule.
\\[-2mm]

\section{Introduction\label{sec:intro}}

The Pitman--Yor process defines a rich and flexible class of random probability measures used as prior distribution in Bayesian nonparametric inference. It originates from the work of \citet{Per:Pit:Yor:92}, further investigated in \citet{Pit:95,Pit:Yor:97}, and

its use in nonparametric inference was initiated by \citet{Ish:Jam:01}. Thanks to its analytical tractability and flexibility, it has found applications in a variety of inferential problems which include species sampling \citep{Lij:etal:07,Fav:etal:09,Nav:etal:08}, survival analysis and gene networks \citep{Jar:etal:10, yang}, linguistics and image segmentation \citep{Teh:06,Sud:Jor:09}, curve estimation \citep{Can:etal:17} and time-series and econometrics \citep{caron,leisen}.

The Pitman--Yor process is a discrete probability measure
\begin{equation}\label{eq:discrete}
  P(\ddr x)=\sum_{i\geq1}p_{i}\delta_{\xi_{i}}(\ddr x)
\end{equation}
where $(\xi_{i})_{i\geq1}$ are \gls{iid} random variables with common distribution $P_0$ on a

Polish space ${\cal X}$, and $(p_{i})_{i\geq1}$ are random frequencies, i.e. $p_i\geq 0$ and $\sum_{i\geq1}p_{i}=1$, independent of $(\xi_{i})_{i\geq1}$. The distribution of the frequencies of the Pitman--Yor process is known in the literature as the two-parameter Poisson--Dirichlet distribution.

Its distinctive property is that the frequencies in {\it size-biased order}, that is the random arrangement in the order of appearance in a simple random sampling without replacement, admit the \textit{stick-breaking representation}, or residual allocation model,

\begin{equation}\label{eq:stick-breaking}
  \p_i\overset{d}{=}V_i\prod_{j=1}^{i-1}(1-V_j),
  \quad V_j\overset{\mbox{\footnotesize ind}}{\sim}
  \mbox{beta}(1-\alpha,\theta+j\alpha)
\end{equation}
for $0\leq \alpha<1$ and $\theta>-\alpha$, see \citet{Pit:Yor:97}. By setting $\alpha=0$ one recovers the Dirichlet process of \citet{Fer:73}.

Representation \eqref{eq:stick-breaking} turns out very useful in devising finite support approximation to the Pitman--Yor process obtained by  truncating the summation in~\eqref{eq:discrete}. A general method consists in setting the truncation level $n$ by replacing $p_{n+1}$ with $1-(p_1+\cdots+p_n)$ in~\eqref{eq:discrete}. The key quantity is the \textit{truncation error} of the infinite summation~\eqref{eq:discrete},
\begin{equation}\label{eq:R_n}
  R_n= \sum_{i>n} \p_i

  = \prod_{j\leq n} (1-V_j).
\end{equation}
since the resulting truncated process, say $P_n(\cdot)$, will be close to $P(\cdot)$ according to
  $|P(A)-P_n(A)|\leq R_n$
for any measurable $A\subset{\cal X}$. It is then important to study the distribution of the truncation error $R_n$ as $n$ gets large in order to control the approximation error. \citet{Ish:Jam:01} proposes to determine the truncation level based on the moments of $R_n$. Cf. also \citet{Ish:Zar:02,Gel:Kot:02}. In this paper we propose and investigate a random truncation by setting $n$ such that $R_n$ is smaller than a predetermined value $\epsilon\in(0,1)$ with probability one. Specifically, we define
\begin{equation}\label{eq:tau_eps}
  \tau(\epsilon)=\min\{n\geq 1:\
  R_n<\epsilon\}

\end{equation}
as the stopping time of the multiplicative process $(R_n)_{n\geq 1}$ and,

following  \citet[Section 4.3.3]{Gho:vdV:17}, we call {\it $\epsilon$-Pitman--Yor ($\epsilon$-PY) process} the Pitman--Yor process truncated at $n=\tau(\epsilon)$, namely
\begin{equation}\label{eq:eps-PY}
  P_\epsilon(\ddr x)
  =\sum_{i=1}^{\tau(\epsilon)}\p_i
  \delta_{\xi_i}(\ddr x)+ R_{\tau(\epsilon)}
  \delta_{\xi_0}(\ddr x),
\end{equation}
where $\xi_0$ has distribution $P_0$, independent of the sequences $(\p_i)_{i\geq 1}$ and $(\xi_i)_{i\geq 1}$.

By construction, $P_\epsilon$ is the finite stick-breaking approximation to $P$ with the smallest number of support points given a predetermined approximation level. In fact $\tau(\epsilon)$ controls the error of approximation according to the total variation bound

\begin{equation}\label{eq:TV}
  d_{TV}(P_\epsilon,P)
  =\sup_{A\subset{\cal X}}|P(A)-P_\epsilon(A)|
  \leq \epsilon
\end{equation}
almost surely (a.s.). As such, it also guarantees the almost sure convergence of measurable functionals of $P$ by the corresponding functionals of $P_\epsilon$ as $\epsilon\to 0$, cf. \citet[Proposition 4.20]{Gho:vdV:17}. A typical application is in Bayesian nonparametric inference on mixture models where the Pitman--Yor process is used as prior distribution on the mixing measure. The approximation $P_\epsilon$ can be applied to the posterior distribution given the latent variables, cf. Section \ref{sec:2.2} for details.

In the Dirichlet process case, $P_\epsilon$ has been studied by \citet{Mul:Tar:98}. In this setting $\tau(\epsilon)-1$ is Poisson distributed with parameter $\theta\log 1/\epsilon$, which makes an exact sampling of the $\epsilon$-approximation \eqref{eq:eps-PY} feasible. This has been implemented in the highly popular R software DPpackage, see \citep{Jar:07,Jar:etal:11}, to draw posterior inference on the random effect distribution of linear and generalized linear mixed effect model.

The main theoretical contribution of this paper is the derivation of the asymptotic distribution of $\tau(\epsilon)$ as $\epsilon\to 0$ for $\alpha>0$.

As \eqref{eq:tau_eps} suggests, the asymptotic distribution of $\tau(\epsilon)$ is related to that of $R_n$ in \eqref{eq:R_n} as $n\to\infty$. According to \citet[Lemma 3.11]{Pit:06}, the latter involves a polynomially tilted stable random variable $T_{\alpha,\theta}$, see Section \ref{sec:2} for a formal definition. The main idea is to work with $T_n=-\log R_n$ so to deal with sums of the independent random variables $Y_i=-\log(1-V_i)$. The distribution of  $\tau(\epsilon)$ can be then studied in terms of the allied {\it renewal counting process}
  $N(t)=\max\{n:\ T_n\leq t \}$,
according to the relation
  $\tau(\epsilon)=N(\log 1/\epsilon)+1$.
The problem boils down to the derivation an appropriate a.s. convergence of $N(t)$ as $t\to\infty$, which, in turn, is obtained from the asymptotic distribution of $T_n$ by showing that $N(t)\to\infty$ a.s. as $t\to\infty$ together with a (non standard) application of the law of large numbers for randomly indexed sequences. This strategy proves successful in establishing the almost sure convergence of $\tau(\epsilon)-1$ to
  $(\epsilon T_{\alpha,\theta}/\alpha)^{-\alpha/(1-\alpha)}$
as $\epsilon\to 0$.

The form of the asymptotic distribution reveals how large the truncation point $\tau(\epsilon)$ is as $\epsilon$ gets small in terms of the model parameters $\alpha$ and $\theta$.

In particular, it highlights the power law behavior of $\tau(\epsilon)$ as $\epsilon\to 0$, namely the growth at the polynomial rate $1/\epsilon^{\alpha/(1-\alpha)}$ compared to the slower logarithmic rate $\theta\log 1/\epsilon$ in the Dirichlet process case. This is further illustrated by a simulation study in which we generate from the asymptotic distribution of $\tau(\epsilon)$ by using Zoltarev's integral representation of the positive stable distribution as in \citet{Dev:09}.

As far as simulation of the $\epsilon$-PY process is concerned, exact sampling is feasible by implementing the stopping rule in \eqref{eq:tau_eps}, that is by simulating the stick breaking frequencies $p_j$ until the error $R_n$ crosses the approximation level $\epsilon$. As this can be computationally expensive when $\epsilon$ is small, as an alternative we propose to use the asymptotic distribution of $\tau(\epsilon)$ by simulating the truncation point first, then run the stick breaking procedure up to that point. It results in an approximate sampler of the $\epsilon$-PY process that we compare with the exact sampler in a simulation study involving moments and mean functionals.

The rest of the paper is organized as follows. In Section~\ref{sec:2}, we derive the asymptotic distribution of $\tau(\epsilon)$ and explain how to use it to simulate from the $\epsilon$-PY process. Section~\ref{sec:3} reports a simulation study on the  distribution of $\tau(\epsilon)$ and on functionals of the $\epsilon$-PY process. In Section \ref{sec:4}, to help the understanding and gain additional insight on the asymptotic distribution, we highlight the connections of $\tau(\epsilon)$ with PitmanÕs theory on random partition structures.

Finally, in Section \ref{appendix:devroye}, we describe  Devroye's algorithm for generating from a polynomially tilted positive stable random variable.

\section{Theory and algorithms}\label{sec:2}

\subsection{Asymptotic distribution of $\tau(\epsilon)$}

In this section we derive the asymptotic distribution of the stopping time $\tau(\epsilon)$ and show how to simulate from it.

We start by introducing the renewal process interpretation which is crucial for the asymptotic results. As explained in the previous section, in order to study the distribution of $\tau(\epsilon)$ it is convenient to work with the log transformation of the truncation error $R_n$ in \eqref{eq:R_n}, that is
\begin{equation}\label{eq:T_n}
  T_n=\sum_{i=1}^n Y_i,\quad
  Y_i=-\log(1-V_i),
\end{equation}
with
  $V_j\overset{\mbox{\footnotesize ind}}{\sim}\mbox{beta}(1-\alpha,\theta+j\alpha)$
as in \eqref{eq:stick-breaking}. Being a sum of  independent and nonnegative random variables, $(T_n)_{n\geq 1}$ takes the interpretation of a (generalized) renewal process with independent waiting times $Y_i$. For $t\geq 0$ define
\begin{equation}\label{eq:N(t)}
  N(t)=\max\{n:\ T_n\leq t \},
\end{equation}
to be the {\it renewal counting process} associated to $(T_n)_{n\geq 1}$, which is related to $\tau(\epsilon)$ via
  $\tau(\epsilon)=N(\log 1/\epsilon)+1$.
Classical renewal theory pertains to iid waiting times while here there is no identity in distribution unless $\alpha=0$, i.e. the Dirichlet process case.

In the latter setting, one gets $Y_i\overset{\mbox{\footnotesize \gls{iid}}}{\sim}\mbox{Exp}(\theta)$ so that $T_n$ has gamma distribution with scale parameter $n$. We immediately get from the relation
  $\{T_n\leq t\}=\{N(t)\geq n\}$
that $N(t)\sim \mbox{Pois}(\theta t)$ and, in turn, that $\tau(\epsilon)-1$ has $\mbox{Pois}(\theta\log(1/\epsilon))$ distribution. As far as asymptotics is concerned, $T_n$ satisfies the \gls{CLT} with
  $(T_n-n/\theta)/(\sqrt{n}/\theta)
  \to_d Z$
where $Z\sim\mbox{N}(0,1)$. The asymptotic distribution of $N(t)$ can be obtained via Ascombe theorem, cf. \citet[Theorem 7.4.1]{Gut:13}, to get
  $(N(t)-\theta t)/(\sqrt{\theta t})
  \rightarrow_d Z$,
as $t\to\infty$, in accordance with the standard normal approximation of the Poisson distribution with large rate parameter.

In the general Pitman--Yor case $\alpha>0$, the waiting times $Y_i$ are no more identically distributed.

More importantly, generalizations of the CLT such as the Lindeberg--Feller theorem

do not apply for $T_n$,

hence we cannot resort to Anscombe's theorem to derive the asymptotic distribution of $N(t)$ and, in turn, of $\tau(\epsilon)$.
Nevertheless, the limit exists but is not normal as stated in Theorem \ref{th:1} below.

To this aim, let $T_\alpha$ be a positive

stable random variable with exponent $\alpha$, that is $\E (\edr^{-s T_\alpha})=\edr^{-s^\alpha}$, and denote its density by $f_\alpha(t)$. A polynomially tilted version of $T_\alpha$ is defined as the random variable $T_{\alpha,\theta}$ with density proportional to $t^{-\theta} f_\alpha(t)$, that is
\begin{equation}\label{eq:density}
  f_{\alpha,\theta}(t)
  = \frac{\Gamma(\theta+1)}{\Gamma(\theta/\alpha+1)}
  t^{-\theta}f_\alpha(t),\quad t> 0.
\end{equation}
The random variable $T_{\alpha,\theta}$ is of paramount importance in the theory of random partition structures associated to the frequency distribution of the Pitman--Yor process, see Section \ref{sec:4} for details. In particular, the convergence of $R_n$ can be express in terms of $T_{\alpha,\theta}$. In Theorem \ref{th:1}

the a.s. limit of $\log N(t)$ as $t\to\infty$ is obtained from that of $T_n=-\log R_n$ as $n\to\infty$ by showing that $N(t)\to\infty$ a.s. as $t\to\infty$ and by an application of the law of large numbers for randomly indexed sequences.

\begin{theorem}\label{th:1}
Let $N(t)$ be defined in \eqref{eq:T_n}--\eqref{eq:N(t)} and $T_{\alpha,\theta}$ be the random variable with density in \eqref{eq:density}. Then
  $t-(1/\alpha-1)\log N(t)+\log\alpha
  \to_{a.s.} \log T_{\alpha,\theta}$
as $t\to\infty$.

\end{theorem}

\begin{proof}
By definition \eqref{eq:N(t)}, the renewal process $N(t)$ is related to the sequence of renewal epochs $T_n$ through
\begin{equation}\label{eq:fundamental}
  \{T_n\leq t\}=\{N(t)\geq n\}.
\end{equation}
Since $N(T_n)=n$, we have $T_{N(t)}=T_n$ when $t=T_n$, thus

  $0=t-T_{N(t)}$
for $t=T_n$.
Moreover, since $N(t)$ is increasing, when $T_n<t<T_{n+1}$, $N(T_n)<N(t)<N(t)+1$, hence
  $T_{N(t)}<t<T_{N(t)+1}$,
i.e.
  $0<t-T_{N(t)}<T_{N(t)+1}-T_{N(t)}=Y_{N(t)+1}$.
Together the two relations above yield
\begin{equation}\label{eq:sandwich}

  0\leq t-T_{N(t)}<Y_{N(t)+1}
\end{equation}
From Lemma 3.11 of \citet{Pit:06} and an application of the continuous mapping theorem \citep[see Theorem 10.1 in][]{Gut:13} the asymptotic distribution of $T_n$ is obtained as
  $$ T_n-(1/\alpha-1)\log n+\log\alpha
  \to_{a.s.} \log T_{\alpha,\theta}
  \quad\mbox{as }n\to\infty.$$
Now we would like to take the limit with respect to $n=N(t)$ as $t\to \infty$, that is apply the law of large numbers for randomly indexed sequence \citet[see Theorem 6.8.1]{Gut:13}. To this aim, we first need to prove that
  $N(t)\to_{a.s.}\infty$
as $t\to\infty$.
Since $N(t)$ is non decreasing, by an application of Theorem 5.3.5 in \citet{Gut:13}, it is sufficient to prove that $N(t)\to\infty$ in probability as $t\to\infty$, that is
  $\Prob(N(t)\geq n)\to 1$
as $t\to\infty$
for any $n\in\N$. But this is an immediate consequence of  the inversion formula \eqref{eq:fundamental}. We have then established that
  $$T_{N(t)}-(1/\alpha-1)\log N(t)+\log\alpha
  \to_{a.s.} \log T_{\alpha,\theta}\quad
  \mbox{as }t\to\infty.$$

To conclude the proof, we need to replace $T_{N(t)}$ with $t$ in the limit above. Note that, from \eqref{eq:sandwich},
  $|t-T_{N(t)}|\leq Y_{N(t)+1}$
so it is sufficient to show that the upper bound goes to zero a.s.. Actually, by a second application of Theorem 6.8.1 \citet{Gut:13} it is sufficient to show that
  $Y_n\to_{a.s.} 0$
as $n\to\infty$.
This last result is established as follows. Recall that
  $Y_j=-\log(1-V_j)$ for
  $V_j\overset{\mbox{\footnotesize ind}}{\sim}\mbox{beta}(1-\alpha,\theta+j\alpha)$.
For $\epsilon>0$,
\begin{align*}
  \Prob(Y_n>\epsilon)
  =\Prob(V_n<\edr^{-\epsilon})
  &=\int_0^{\edr^{-\epsilon}}
  \frac{\Gamma(\theta+n\alpha+1-\alpha)}
  {\Gamma(\theta+n\alpha)\Gamma(1-\alpha)}
  v^{\theta+n\alpha-1}(1-v)^{-\alpha}\ddr x\\
  &\leq \frac{(1-\edr^{-\epsilon})^{-\alpha}}
  {\Gamma(1-\alpha)}
  \frac{\Gamma(\theta+n\alpha+1-\alpha)}
  {\Gamma(\theta+n\alpha)}
  \frac{\edr^{-\epsilon(\theta+n\alpha)}}{\theta+n\alpha}\\

  &=\frac{(1-\edr^{-\epsilon})^{-\alpha}}{\Gamma(1-\alpha)}
  (\theta+n\alpha)^{-\alpha}
  \edr^{-\epsilon(\theta+n\alpha)}
  \Big(1+O\Big(\frac{1}{\theta+n\alpha}\Big)\Big)
\end{align*}
where in the last equality we have used Euler's formula

  $$\Gamma(z+\alpha)/\Gamma(z+\beta)=z^{\alpha-\beta}\left[
  1+\frac{(\alpha-\beta)(\alpha+\beta-1)}{2z}+O(z^{-2})\right]$$

for $z\to\infty$, see \citet{Tri:Erd:51}. So we have established that $\sum_{n\geq 1}\Prob(Y_n>\epsilon)<\infty$ and Borel--Cantelli lemma yields the desired a.s. convergence.
\end{proof}

The asymptotic distribution of $\tau(\epsilon)$ is readily derived from Theorem \ref{th:1} via the formula
  $\tau(\epsilon)=N(\log 1/\epsilon)+1$

and an application of the continuous mapping theorem. The proof is omitted.

\begin{theorem}\label{cor:limiting_tau_eps}
Let $\tau(\epsilon)$ be defined in \eqref{eq:tau_eps} and $T_{\alpha,\theta}$ be the random variable with density in \eqref{eq:density}. Then

  $\tau(\epsilon)-1

  \sim_{a.s.}
  (\epsilon T_{\alpha,\theta}/\alpha)^{-\alpha/(1-\alpha)}$
as $\epsilon\to 0$.

\end{theorem}

In order to sample from the asymptotic distribution of $\tau(\epsilon)$, the key ingredient is random generation from the polynomially tilted stable random variable $T_{\alpha,\theta}$.
Following \citet{Dev:09}, we resort to Zoltarev's integral representation, so let $A(u)$ be the Zoltarev function
  $$A(x)=\bigg(
  \frac{\sin(\alpha x)^\alpha\sin((1-\alpha)x)^{1-\alpha}}{\sin(x)}
  \bigg)^{\frac{1}{1-\alpha}},\quad x\in[0,\pi]$$
and $Z_{\alpha,b}$, $\alpha\in(0,1)$ and $b>-1$ be a Zoltarev random variable with density given by
  $$f(x)=\frac{\Gamma(1+b\alpha)\Gamma(1+b(1-\alpha))}
  {\pi\Gamma(1+b)A(x)^{b(1-\alpha)}},\quad 0\leq x\leq \pi$$

According to Theorem 1 of \citet{Dev:09}, for $G_a$ a gamma distributed  random variable with shape $a>0$ and unit rate,
  $$T_{\alpha,\theta}\overset{d}{=}
  \bigg( \frac{A(Z_{\alpha,\theta/\alpha})}
  {G_{1+\theta(1-\alpha)/\alpha}}
  \bigg)^{\frac{1-\alpha}{\alpha}}$$
so that random variate generation simply requires one gamma random variable and one Zolotarev random variable. For the latter, rejection sampler can be used as detailed in \citet{Dev:09}. See {\sc Algorithm 3} in Section \ref{appendix:devroye}.

\subsection{Simulation of $\epsilon$-PY process}
\label{sec:2.2}

Given $\alpha,\theta,\epsilon$ and a probability measure $P_0$ on ${\cal X}$, an $\epsilon$-PY process can be generated by implementing the stopping rule in the definition of $\tau(\epsilon)$, cf. \eqref{eq:tau_eps}. The algorithm consists in a while loop as follows:

\begin{quote}
\centerline{{\sc Algorithm 1} (Exact sampler of $\epsilon$-PY)}
\vspace{4pt}

{\tt
1. set $i= 1$, $R= 1$
\vspace{4pt}

2. while $R\geq\epsilon$:$\quad$ generate $V$

  from $\text{\rm beta}(1-\alpha,\theta+i\alpha)$.
\\
\phantom{3. while $R>\epsilon$:$\quad$} set
  $p_i= VR$, $R= R(1-V)$, $i= i+1$
\vspace{4pt}

3. set
  $\tau= i$, $R_\tau=R$
\vspace{4pt}

4. generate $\tau+1$ random variates $\xi_{0},\xi_{1},\ldots,\xi_{\tau}$ from $P_0$
\vspace{4pt}

5. set
  $  P_\epsilon(\ddr x)
  =\sum_{i=1}^{\tau}\p_i
  \delta_{\xi_i}(\ddr x)+ R_\tau
  \delta_{\xi_0}(\ddr x)$
}
\end{quote}

When $\epsilon$ is small, the while loop happens to be computationally expensive since conditional evaluations at each iteration slow down computation, and memory allocation for the frequency and location vectors cannot be decided beforehand.
In order to avoids these pitfalls and make the algorithm faster, one should generate the stopping time $\tau(\epsilon)$ first, and the frequencies up to that point later. We propose to exploits the asymptotic distribution of $\tau(\epsilon)$ in Theorem~\ref{cor:limiting_tau_eps}

as follows:

\begin{quote}
\centerline{{\sc Algorithm 2} (Approximate sampler of $\epsilon$-PY)}
\vspace{4pt}

{\tt
1: generate $T\overset{d}{=} T_{\alpha,\theta}$
\vspace{4pt}

2: set
  $\tau\leftarrow 1+\lfloor
  (\epsilon T/\alpha)^{-\alpha/(1-\alpha)}
  \rfloor$
\vspace{4pt}

3. for $i=1,\ldots,\tau$:$\quad$ generate $V_i$
  from $\text{\rm beta}(1-\alpha,\theta+i\alpha)$.
\\
\phantom{3. for $i=1,\ldots,\tau$:$\quad$} set
  $p_i= V_i\prod_{j=1}^{i-1}(1-V_j)$
\vspace{4pt}

4. set
  $R_\tau= 1-\sum_{i=1}^\tau p_i=\prod_{i=1}^\tau (1-V_j)$
\vspace{4pt}

5: generate $\tau+1$ random variates $\xi_{0},\xi_{1},\ldots,\xi_{\tau}$ from $P_0$
\vspace{4pt}

6: set
  $  P_\epsilon(\ddr x)
  =\sum_{i=1}^{\tau}\p_i
  \delta_{\xi_i}(\ddr x)+ R
  \delta_{\xi_0}(\ddr x)$
}
\end{quote}

{\sc Algorithm 2} is an {\it approximate} sampler of the $\epsilon$-PY process (while {\sc Algorithm 1} is an {\it exact} one) since it introduces two sources of approximations. First, through the use of the asymptotic distribution of $\tau(\epsilon)$. Second, through {\tt Step 3} since the $V_i$'s are not generated according to the conditional distribution given $\tau(\epsilon)=n+1$ (or, equivalently $R_n<\epsilon$), rather unconditionally. Finding the conditional distribution of $V_i$, or an asymptotic approximation thereof, is not an easy task and is object of current research. In terms of the renewal process in \eqref{eq:T_n}, the problem is to generate the waiting time $Y_i=-\log(1-V_i)$, $i=1,\ldots,n$, from the conditional distribution of the renewal epochs $(T_1,\ldots,T_n)$ given $T_n\leq t=-\log 1/\epsilon$. The latter does not seem to be easily tractable, as it would be if $N(t)$ were a mixed sample processes or, equivalently, a Cox process, cf. \citet[Section 6.3]{Gra:97}.

A typical use of samples from the Pitman--Yor process we have in mind is in infinite mixture models. In fact, the discrete nature of the Pitman--Yor process makes it a suitable prior on the \textit{mixing distribution}. {\sc Algorithm 1} or 2 can be then applied to approximate a functional of the posterior distribution of the mixing distribution. In such models, the process components can be seen as latent features exhibited by the data.

Let $P$ denote such a process, $\nsample$ denote the sample size and  $X_{1:\nsample}=(X_1,\ldots,X_\nsample)$ be exchangeable sequence from $P$, that is $X_{1:\nsample} | P \overset{\mbox{\footnotesize \gls{iid}}}{\sim} P$. $X_{1:\nsample}$ are latent variables in a model conditionally on which observed data $Y_{1:\nsample}$ come from: $Y_j | X_j \overset{\mbox{\footnotesize ind}}{\sim} f(\,\cdot\, | X_j)$

where $f$ denotes a kernel density.

Actually, independence is not necessary here and applications also encompass dependent models such as Markov chain transition density estimation.

In order to deal with the infinite dimensionality of the process, one strategy is to marginalizes it and draws posterior inference with a \textit{marginal} sampler.

Since draws from a marginal sampler allows to make inference only on posterior expectations of the process, for more general functionals of $P$, in the form of $\psi(P)$, one typically needs to resort to an additional sampling step. Exploiting the composition rule
  $\law(\psi(P)|\data)
  = \law(\psi(P)|\latent)\times\law(\latent|\data)$
this additional step boils down to sampling $P$ conditional on latent variables $\latent$. At this stage, recalling the conditional conjugacy of the Pitman--Yor process is useful. Among $\latent$, there are a number $k\leq \nsample$ of unique values that we denote by $\latentunique$. Let $\frequencies$ denote their frequencies. Then the following identity in distribution holds

  $$P|\latent = \sum_{j=1}^{k}q_j \delta_{X_j^*} + q_{k+1}P^*,$$

where, independently, $(q_1,\ldots,q_k,q_{k+1})\sim\mbox{Dirichlet}(\nsample_1^*-\alpha,\ldots,\nsample_k^*-\alpha,\theta+\alpha k)$ and $P^*$ is a Pitman--Yor process of parameter $(\alpha,\theta+\alpha k)$, see Corollary 20 of \citet{Pit:96a}. Thus sampling from $\law(P|\latent)$, hence from $\law(\psi(P)|\latent)$, requires sampling the infinite dimensional $P^*$.  Cf. \citet[Section 4.4]{Ish:Jam:01}. For the sake of comparison, the conjugacy of the Dirichlet process similarly leads to the need of sampling an infinite dimensional process, where $P|X_{1:\nsample}$ takes the form of a Dirichlet process. As already noticed, the truncation of the Dirichlet process is very well understood, both theoretically and practically. The popular \textsf{R} package \textsf{DPpackage} \citep{Jar:07,Jar:etal:11} makes use of the \textit{posterior} truncation point $\tau^*(\epsilon)$, as defined in \eqref{eq:eps-PY}, but here with respect to the posterior distribution of the process. Thus, it satisfies

  $\tau^*(\epsilon)-1\sim \mbox{Pois}((\theta+\nsample)
  \log(1/\epsilon)),$

where $\theta +\nsample$ is the precision of the posterior Dirichlet process.
Adopting here similar lines for the Pitman--Yor process, we replace $P^*$ by the truncated process $P^*_\epsilon$
\begin{equation*}
  P^*_\epsilon(\ddr x)
  =\sum_{i=1}^{\tau^*(\epsilon)}\p^*_i
  \delta_{\xi_i}(\ddr x)+ R_{\tau^*(\epsilon)}
  \delta_{\xi_0}(\ddr x),
\end{equation*}
cf. equation \eqref{eq:eps-PY}. Here $(\p_i^*)_{i\geq 1}$ are defined according to \eqref{eq:stick-breaking} with  $\theta+\alpha k$ in place of $\theta$, i.e.
  $V_j\overset{\mbox{\footnotesize ind}}{\sim}
  \text{beta}(1-\alpha,\theta+\alpha (k+j))$.

Hence, according to Theorem~\ref{cor:limiting_tau_eps} we have

\begin{equation*}
  \tau^*(\epsilon)-1\sim_{a.s.}
  (\epsilon T_{\alpha,\theta+\alpha k}/\alpha)^{-\alpha/(1-\alpha)},
  \quad\mbox{as }\epsilon\to 0
\end{equation*}

hence {\sc Algorithm 2} can be applied here.

\section{Simulation study}\label{sec:3}

\subsection{Stopping time $\tau(\epsilon)$}\label{sec:3.1}

According to Theorem \ref{cor:limiting_tau_eps}, the asymptotic distribution of $\tau(\epsilon)$ changes with $\epsilon$,  $\alpha$ and $\theta$.

For illustration, we simulate $\tau(\epsilon)$ from {\tt Steps 1.-2.} in {\sc Algorithm 2} using Devroye's sampler, cf. {\sc Algorithm 3} in Section \ref{appendix:devroye}. In Figure \ref{fig:asymp-approx} we compare density plots obtained with $10^4$ iterations with respect to different combinations of $\epsilon$, $\alpha$ and $\theta$.

The plot in the left panel shows how smaller values of $\epsilon$ result in larger values of $\tau(\epsilon)$. In fact, as $\epsilon\to 0$, $\tau(\epsilon)$ increases proportional to $1/\epsilon^{\alpha/(1-\alpha)}$. Note also that $(\epsilon T_{\alpha,\theta}/\alpha)^{-\alpha/(1-\alpha)}$ is nonnegative for $T_{\alpha,\epsilon}<\alpha/\epsilon$, which happens with high probability when $\epsilon$ is small. As for $\alpha$, the plot in the central panel shows how $\tau(\epsilon)$ increases as $\alpha$ gets large. In fact, it is easy to see that $(\epsilon T_{\alpha,\theta}/\alpha)^{-\alpha/(1-\alpha)}$ is increasing in $\alpha$ when $T_{\alpha,\epsilon}<\edr^{1-\alpha}\alpha/\epsilon$, which also happens with high probability when $\epsilon$ is small, so the larger $\alpha$, the more stick-breaking frequencies are needed in order to account for a prescribed approximation error $\epsilon$. Finally, the plot in the right panel shows that the larger $\theta$, the larger $\tau(\epsilon)$. In fact, by definition, the polynomial tilting makes $T_{\alpha,\theta}$ stochastically decreasing in $\theta$.

\begin{figure}[!ht]
\centering

\begin{tabular}{ccc}
\includegraphics[width = .33\textwidth]{epsilon_varying-1.pdf}
\includegraphics[width = .33\textwidth]{alpha_varying-1.pdf}&
\includegraphics[width = .33\textwidth]{theta_varying-1.pdf}
\end{tabular}
\caption{
Density plot for the asymptotic approximation of $\tau(\epsilon)$ based on $10^4$ values under the following parameter configurations.
Left: $\epsilon \in\{0.10, 0.05, 0.01\}$, $\alpha = 0.4$, $\theta = 1$.
Center: $\alpha \in\{0.4, 0.5, 0.6\}$, $\theta = 1$, $\epsilon = 0.1$.
Right: $\theta \in\{0,1,10\}$, $\alpha = 0.25$, $\epsilon = 0.05$.

}
\label{fig:asymp-approx}
\end{figure}

In order to illustrate the rate of convergence in Theorem \ref{cor:limiting_tau_eps}, we compare next the exact distribution of $\tau(\epsilon)$ with the asymptotic one. To do so, we repeat the following experiment several time: we simulate $\tau(\epsilon)$ from {\tt Steps 1.-3.} in {\sc Algorithm 1}, then we compare the empirical distribution of
  $(\epsilon/\alpha)^{\alpha}
  (\tau(\epsilon)-1)^{1-\alpha}$
with
  $T_{\alpha,\theta}^{-\alpha}$,
the latter corresponding to the $\alpha$-diversity of the PY process, see Section \ref{sec:4} for a formal definition. In Table \ref{tab:diversity} we report the Kolmogorov distance together with expected value, median, first and third quartiles for both the exact and the asymptotic distribution obtained with $10^4$ iterations. This is repeated for $\alpha=0.5$, $\theta=\{0,1,10\}$ and $\epsilon=\{0.10,0.05,0.01\}$. As expected, as we decrease $\epsilon$, the Kolmogorov distance gets smaller to somehow different rates according to the parameter choice. The derivation of convergence rates is left for future research.

\begin{table}[!ht]
\setlength{\tabcolsep}{3.5pt}

\vspace*{4pt}
\begin{center}
\begin{tabular}{cc@{\hskip 15pt}r@{\hskip 15pt}rr@{\hskip 15pt}rr@{\hskip 15pt}rr@{\hskip 15pt}rr}
\hline
  &&{\boldmath${d_K}$}& \multicolumn{2}{c}{\textbf{Mean}} & \multicolumn{2}{c}{\boldmath$25\%$} &
\multicolumn{2}{c}{\textbf{Median}} & \multicolumn{2}{c}{\boldmath$75\%$} \\
 $\theta$ & $\epsilon$ &  & As & Ex & As & Ex & As & Ex & As & Ex \\
\hline
0 & 0.10 & 3.42 & 1.06 & 1.05 & 0.45 & 0.45 & 0.89 & 0.89 & 1.61 & 1.55 \\
  0 & 0.05 & 2.17 & 1.10 & 1.08 & 0.45 & 0.45 & 0.95 & 0.95 & 1.64 & 1.58 \\
  0 & 0.01 & 1.73 & 1.14 & 1.11 & 0.45 & 0.45 & 0.97 & 0.95 & 1.64 & 1.60 \\
  1 & 0.10 & 4.79 & 2.24 & 2.14 & 1.55 & 1.48 & 2.14 & 2.10 & 2.86 & 2.76 \\
  1 & 0.05 & 2.38 & 2.25 & 2.20 & 1.55 & 1.52 & 2.17 & 2.14 & 2.86 & 2.79 \\
  1 & 0.01 & 1.40 & 2.26 & 2.25 & 1.57 & 1.54 & 2.19 & 2.19 & 2.87 & 2.85 \\
  10 & 0.10 & 11.93 & 6.39 & 6.07 & 5.69 & 5.40 & 6.34 & 6.06 & 7.04 & 6.72 \\
  10 & 0.05 & 6.12 & 6.39 & 6.24 & 5.70 & 5.56 & 6.34 & 6.22 & 7.05 & 6.88 \\
  10 & 0.01 & 1.93 & 6.40 & 6.37 & 5.71 & 5.70 & 6.34 & 6.34 & 7.05 & 7.00
\\
\hline
\end{tabular}
\end{center}
\caption{
Summary statistics for the asymptotic distribution (As)  and exact distribution (Ex) of $\tau(\epsilon)$ at the scale of the $\alpha$-diversity based on $10^4$ values. The Kolmogorov distance ($d_K$) is between the empirical cumulative distribution function of the sample from the exact distribution and the asymptotic one (multiplied by a factor of 100). The parameter values are $\alpha = 0.5$, $\theta\in\{0,1,10\}$ and $\epsilon\in\{0.10,0.05,0.01\}$.
}\label{tab:diversity}
\end{table}

\subsection{Functionals of the $\epsilon$-PY process}\label{sec:3.2}

In the case that $P$ is defined on $\mathcal{X}\subseteq\mathds{R}$, the total variation bound \eqref{eq:TV} implies that $|F(x)-F_\epsilon(x)|<\epsilon$ almost surely for any $x\in\mathds{R}$, where $F_\epsilon$ and $F$ are the cumulative distribution functions of $P_\epsilon$ and $P$.

Also, measurable functionals $\psi(P)$ such as the mean $\mu=\int x P(\ddr x)$ can be approximated in distribution by the corresponding functionals $\psi(P_\epsilon)$.

For illustration, we set $\mathcal{X}=[0,1]$ and $P_0$ the uniform distribution on $[0,1]$. For given $\alpha$ and $\theta$, we then compare the distribution under $P$ with that under the $\epsilon$-PY process $P_\epsilon$ for $F(1/2)$, $F(1/3)$ and $\mu=\int x P(\ddr x)$.

As for the distribution of the finite dimensional distributions $F(1/2)$ and $F(1/3)$ under the full process $P$, we set $\alpha=0.5$ so to exploit  results in \citet{Jam:etal:10}. According to their Proposition 4.7,  the finite dimensional distributions of $P$ when $\alpha=0.5$ are given by
  $$f(w_1,\ldots,w_{n-1})=
  \frac{(\prod_{i=1}^np_i)\Gamma(\theta+n/2)}
  {\pi^{(n-1)/2}\Gamma(\theta+1/2)}
  \frac{w_1^{-3/2}\cdots w_{n-1}^{-3/2}
  (1-\sum_{i=1}^{n-1}w_i)^{-3/2}}
  {\mathcal{A}_n(w_1,\ldots,w_{n-1})^{\theta+n/2}}$$
for any partition $A_1,\ldots,A_n\in\mathcal{X}$ with $p_i=P_0(A_i)$ and
  $\mathcal{A}_n(w_1,\ldots,w_{n-1})
  =\sum_{i=1}^np_i^2 w_i^{-1}
  +p_n^2 (1-\sum_{i=1}^{n-1}w_i)^{-1}$.
Direct calculation shows that $F(1/2)$ has beta distribution with parameters $(\theta+1/2,\theta+1/2)$ while $F(1/3)$ has density
  $$f(w)=\frac{2}{\sqrt\pi}9^\theta
  \frac{\Gamma(\theta+1)}{\Gamma(\theta+1/2)}
  \frac{(w(1-w))^{\theta-1/2}}{(1+3w)^{\theta+1}}.$$
As for the mean functional $\mu=\int xP(\ddr x)$, the distribution under the full process $P$ is approximated by simulations by setting a deterministic truncation point sufficiently large. As for the distribution under $P_\epsilon$, we use both {\sc Algorithm 1} and {\sc Algorithm 2}.

In Figure \ref{fig:random-probabilities} we compare the density plots of $F(1/2)$ for $\epsilon=\{0,1.0.05,0.001\}$ and $\theta=\{0,10\}$ under $P_\epsilon$ with the beta density under $P$ so to illustrate that the two distributions gets close as $\epsilon$ gets small.

\begin{figure}[!ht]

\centering
\begin{tabular}{c}
\includegraphics[width = \textwidth]{median-1.pdf}\\
\includegraphics[width = \textwidth]{median-2.pdf}
\end{tabular}

\caption{Density plots for the random probability $F(1/2)$ using the {\sc Algorithm 2} (in red solid curve) and {\sc Algorithm 1} (in blue dashed curve) to sample from the $\epsilon$-PY process. The density under the Pitman--Yor process is the black dotted curve. The parameter $\alpha$ is fixed equal to $0.5$, $\theta$ is equal to $0$ on the first row and $10$ on the second row, while $\epsilon$ is respectively equal to $\{0.10, 0.05, 0.01\}$ in the left, center and right columns.}
\label{fig:random-probabilities}
\end{figure}

As for $F(1/3)$ and $\mu = \int x P(\ddr x)$, in Tables \ref{tab:F13} and \ref{tab:Fmu} we report the Kolmogorov distance between $P$ and $P_\epsilon$ for the two sampling algorithms, together with expected value, median, first and third quartiles. For each case and each parameter configuration, we have sampled $10^4$ trajectories from the $\epsilon$-PY process and $10^4$ trajectories from the Pitman--Yor process in the case of $\mu = \int x P(\ddr x)$. As expected, the Kolmogorov distances are generally larger, still close, when using {\sc Algorithm 2} versus {\sc Algorithm 1} due to the approximate nature of the former.

\begin{table}[!ht]
\def~{\hphantom{0}}
\caption{Simulation study on $F(1/3)$}
\label{tab:F13}
\setlength{\tabcolsep}{2pt}
\vspace*{4pt}
\begin{center}
\begin{tabular}{cc@{\hskip 8pt}rr@{\hskip 8pt}rrr@{\hskip 8pt}rrr@{\hskip 8pt}rrr@{\hskip 8pt}rrr}
\hline
  & & \multicolumn{2}{c}{\boldmath${d_K}$} & \multicolumn{3}{c}{\textbf{Mean}} & \multicolumn{3}{c}{\boldmath$25\%$} &
\multicolumn{3}{c}{\textbf{Median}} & \multicolumn{3}{c}{\boldmath$75\%$} \\
$\theta$ & $\epsilon$ &
{\sc Al1} & {\sc Al2} &
{\sc Al1} & {\sc Al2} & {\sc PY} &
{\sc Al1} & {\sc Al2} & {\sc PY} &
{\sc Al1} & {\sc Al2} & {\sc PY} &
{\sc Al1} & {\sc Al2} & {\sc PY} \\
\hline
0 & 0.10 & 16.29 & 16.48 & 0.33 & 0.33 & 0.33 & 0.04 & 0.01 & 0.04 & 0.20 & 0.16 & 0.20 & 0.60 & 0.64 & 0.59 \\
0 & 0.05 & 11.53 & 12.52 & 0.33 & 0.33 & 0.33 & 0.05 & 0.01 & 0.04 & 0.20 & 0.17 & 0.20 & 0.58 & 0.63 & 0.59 \\
  0 & 0.01 & 5.49 & 5.60 & 0.34 & 0.33 & 0.33 & 0.04 & 0.03 & 0.04 & 0.21 & 0.19 & 0.20 & 0.59 & 0.61 & 0.59 \\
  1 & 0.10 & 3.08 & 5.65 & 0.33 & 0.33 & 0.33 & 0.14 & 0.12 & 0.14 & 0.29 & 0.28 & 0.28 & 0.49 & 0.50 & 0.49 \\
  1 & 0.05 & 1.34 & 3.11 & 0.33 & 0.33 & 0.33 & 0.14 & 0.13 & 0.14 & 0.28 & 0.28 & 0.28 & 0.48 & 0.50 & 0.49 \\
  1 & 0.01 & 0.56 & 0.89 & 0.33 & 0.34 & 0.33 & 0.14 & 0.14 & 0.14 & 0.28 & 0.29 & 0.28 & 0.49 & 0.49 & 0.49 \\
  10 & 0.10 & 3.10 & 3.81 & 0.33 & 0.33 & 0.33 & 0.25 & 0.25 & 0.26 & 0.32 & 0.32 & 0.32 & 0.40 & 0.41 & 0.40 \\
  10 & 0.05 & 1.41 & 1.38 & 0.33 & 0.33 & 0.33 & 0.26 & 0.26 & 0.26 & 0.32 & 0.32 & 0.32 & 0.40 & 0.40 & 0.40 \\
  10 & 0.01 & 0.75 & 0.65 & 0.33 & 0.33 & 0.33 & 0.26 & 0.26 & 0.26 & 0.33 & 0.32 & 0.32 & 0.40 & 0.40 & 0.40 \\
\hline
\end{tabular}
\end{center}
\par

\medskip
\caption{Simulation study on $\mu = \int x P(\ddr x)$}
\label{tab:Fmu}
\setlength{\tabcolsep}{2pt}

\begin{center}
\begin{tabular}{cc@{\hskip 8pt}rr@{\hskip 8pt}rrr@{\hskip 8pt}rrr@{\hskip 8pt}rrr@{\hskip 8pt}rrr}
\hline
  & & \multicolumn{2}{c}{\boldmath${d_K}$} & \multicolumn{3}{c}{\textbf{Mean}} & \multicolumn{3}{c}{\boldmath$25\%$} &
\multicolumn{3}{c}{\textbf{Median}} & \multicolumn{3}{c}{\boldmath$75\%$} \\
$\theta$ & $\epsilon$ &
{\sc Al1} & {\sc Al2} &
{\sc Al1} & {\sc Al2} & {\sc PY} &
{\sc Al1} & {\sc Al2} & {\sc PY} &
{\sc Al1} & {\sc Al2} & {\sc PY} &
{\sc Al1} & {\sc Al2} & {\sc PY} \\
\hline
0 & 0.10 & \hphantom{1}1.60 & \hphantom{1}3.57 & 0.50 & 0.50 & 0.50 & 0.36 & 0.34 & 0.36 & 0.50 & 0.50 & 0.50 & 0.64 & 0.67 & 0.65 \\
0 & 0.05 & 0.94 & 2.72 & 0.50 & 0.50 & 0.50 & 0.35 & 0.34 & 0.36 & 0.50 & 0.50 & 0.50 & 0.64 & 0.66 & 0.65 \\
  0 & 0.01 & 1.18 & 2.10 & 0.50 & 0.50 & 0.50 & 0.36 & 0.35 & 0.36 & 0.50 & 0.50 & 0.50 & 0.64 & 0.65 & 0.65 \\
  1 & 0.10 & 1.61 & 3.18 & 0.50 & 0.50 & 0.50 & 0.40 & 0.39 & 0.40 & 0.50 & 0.50 & 0.50 & 0.60 & 0.61 & 0.60 \\
  1 & 0.05 & 1.28 & 2.32 & 0.50 & 0.50 & 0.50 & 0.40 & 0.40 & 0.40 & 0.50 & 0.50 & 0.50 & 0.59 & 0.60 & 0.60 \\
  1 & 0.01 & 1.12 & 0.57 & 0.50 & 0.50 & 0.50 & 0.40 & 0.41 & 0.40 & 0.50 & 0.50 & 0.50 & 0.60 & 0.60 & 0.60 \\
  10 & 0.10 & 2.81 & 4.18 & 0.50 & 0.50 & 0.50 & 0.45 & 0.46 & 0.46 & 0.50 & 0.50 & 0.50 & 0.55 & 0.55 & 0.54 \\
  10 & 0.05 & 1.78 & 1.28 & 0.50 & 0.50 & 0.50 & 0.46 & 0.46 & 0.46 & 0.50 & 0.50 & 0.50 & 0.54 & 0.54 & 0.54 \\
  10 & 0.01 & 2.01 & 1.09 & 0.50 & 0.50 & 0.50 & 0.46 & 0.46 & 0.46 & 0.50 & 0.50 & 0.50 & 0.54 & 0.54 & 0.54 \\
\hline
\end{tabular}
\end{center}
\captionsetup{labelformat=empty}
\caption*{Summary statistics for $F(1/3)$ (Table \ref{tab:F13}) and $\mu = \int x P(\ddr x)$ (Table \ref{tab:Fmu}) using {\sc Algorithm 1} ({\sc Al1}) and {\sc Algorithm 2} ({\sc Al2}) to sample from the $\epsilon$-PY process. The Kolmogorov distance ($d_K$) is  between the cumulative distribution functions with respect to the Pitman--Yor (PY) process (multiplied by a factor of 100). The parameter values are $\alpha = 0.5$, $\theta\in\{0,1,10\}$ and $\epsilon\in\{0.10,0.05,0.01\}$.}
\end{table}
\medskip

\section{Connections with random partition structures}\label{sec:4}

\subsection{$\alpha$-diversity and asymptotic distribution of $R_n$}\label{sec:4.1}

The random variable $T_{\alpha,\theta}$ in Theorem \ref{th:1} plays a key role in the Pitman--Yor process, in particular for its link with the {\it $\alpha$-diversity} of the process. The $\alpha$-diversity is defined as the almost sure limit of $n^{-\alpha}K_n$ where $K_n$ denotes the (random) number of

unique values in the first $n$ terms of an exchangeable sequence from $P$ in \eqref{eq:discrete}. According to Theorem 3.8 in \citet{Pit:06}, $n^{-\alpha}K_n\sim_{a.s.}(T_{\alpha,\theta})^{-\alpha}$, in particular, for $\theta=0$, $T_\alpha^{-\alpha}$ has a Mittag-Leffler distribution with $p$-th moments $\Gamma(p+1)/\Gamma(p\alpha+1)$, $p>-1$. According to \citet[Lemma 3.11, eqn (3.36)]{Pit:06}, the asymptotic distribution of the truncation error $R_n$ can be derived from that of $K_n$ to get
  $R_n\sim_{a.s.}
  \alpha (T_{\alpha,\theta})^{-1}

  \,n^{1-1/\alpha}$
as $n\to\infty$.

The proof relies on Kingman's representation of random partitions \citep{Kin:78} together with techniques set forth by \citet{Gne:Han:Pit:07}. In the proof of Theorem \ref{th:1} the asymptotic distribution of $T_n=-\log R_n$ is a direct consequence of the above by an application of the continuous mapping theorem.

When $\theta=0$ it is possible to give an interpretation of the asymptotic distribution of $R_n$ in terms of the jumps of a stable subordinator. In this case the weights of $P$ can be represented as the renormalized jumps of a stable subordinator, with $T_\alpha$ denoting  the total mass. Denote the jumps as $(J_i)_{i\geq 1}$ in decreasing order and $(\tilde J_i)_{i\geq 1}$ when in size-biased order,
  $$T_\alpha=\sum_{i\geq 1} J_i
  =\sum_{i\geq 1} \tilde J_i,\quad \text{and }
  T_\alpha R_n=\sum_{i>n} \tilde J_i.$$
By the asymptotic distribution of $R_n$
  $n^{1/\alpha-1}
  \sum_{i>n} \tilde J_i\to_{a.s.} \alpha$
as $n\to\infty$,
that is, the small jumps of the stable subordinator (in size-biased random order), interpreted as the \comillas{dust}, once properly renormalized, converge to the \comillas{proportion} $\alpha$, in the same vein as the number of singletons is asymptotically a $\alpha$ proportion of the number of groups in a sample of size $n$, see Lemma 3.11, eqn (3.39), of \citet{Pit:06}.

\subsection{Regenerative random compositions and Anscombe's theorem}\label{sec:4.2}

We review next the connections of the counting renewal process $N(t)$ defined in \eqref{eq:T_n}--\eqref{eq:N(t)} and the theory of regenerative random compositions. The reader is referred to the survey of \citet{Gne:10} for a review. Recall that, when $\alpha=0$ (Dirichlet process case), $V_i\overset{\mbox{\footnotesize \gls{iid}}}{\sim}\text{beta}(1,\theta)$ in the stick-breaking representation \eqref{eq:stick-breaking}, and in turns $Y_i=-\log(1-V_i)\overset{\mbox{\footnotesize \gls{iid}}}{\sim}\mbox{Exp}(\theta)$ and $T_n=-\log R_n\sim\mbox{Gamma}(n,\theta)$. By direct calculus, $N(t)\sim\mbox{Pois}(\theta t)$ so that $\tau(\epsilon)-1=N(\log 1/\epsilon)$ has $\mbox{Pois}(\theta\log 1/\epsilon)$ distribution.
More generally, the stick-breaking frequencies $(\p_i)_{i\geq 1}$ correspond to the gaps in $[0,1]$ identified by the {\it multiplicative regenerative} set ${\cal R}\subset (0,1)$ consisting of the random partial sums $1-R_k=\sum_{i\leq k}p_i$. The complement open set ${\cal R}^c=(0,1)/\cal R$ can be represented as a disjoint union of countably many open intervals or gaps,
  ${\cal R}^c=\bigcup_{k=0}^\infty(1-R_{k},1-R_{k+1})$, $R_0=1$.
A random composition of the integer $n$ into an ordered sequence $\varkappa_n=(n_1,n_2,\ldots,n_k)$ of positive integers with $\sum_{j}n_j=n$ can be generated as follows: independently of ${\cal R}$, sample $U_1,U_2,\ldots$ from the uniform distribution on $[0,1]$ and group them in clusters by the rule: $U_i,U_j$ belong to the same cluster if they hit the same interval. The random composition of $\varkappa_n$ corresponds then to the record of positive counts in the left-to-right order of the intervals. The composition structure $(\varkappa_n)$ is called regenerative since for all $n>m\geq 1$, conditionally given the first part of $\varkappa_n$ is $m$, if the part is deleted then the remaining composition of $n-m$ is distributed like $\varkappa_{n-m}$. The regenerative set ${\cal R}$ corresponds to the closed range of the multiplicative subordinator $\{1-\exp(-S_t),t\geq0\}$, where $S_t$ is the compound Poisson process with L\'evy intensity $\tilde\nu(\ddr y)=\theta\edr^{-\theta y}\ddr y$.

Since the range of $S_t$ is a homogeneous Poisson point process on $\R_+$ with rate $\theta$, ${\cal R}$ is an inhomogeneous Poisson point process ${\cal N}(\ddr x)$ on $[0,1]$ with L\'evy intensity
  $\nu(\ddr x)=\theta/(1-x)\ddr x$
so that, for $t=\log 1/\epsilon$,
  $$N(\log 1/\epsilon)={\cal N} [0,1-\epsilon]
  \sim\mbox{Pois}(\lambda),\quad
  \lambda=\int_0^{1-\epsilon}
  \frac{\theta}{1-x}\ddr x=\theta\log 1/\epsilon$$
as expected. Suppose now that $(V_i)$ are independent copies of some random variable $V$ on $[0,1]$, not necessarily $\text{beta}(1,\theta)$ distributed. The corresponding random composition structure has been studied in \citet{Gne:04,Gne:etal:09} as the outcome of a {\it Bernoulli sieve} procedure. We recall here the relevant asymptotic analysis. Let $\mu=\E(-\log(1-V))$ and $\sigma^2=\Var(-\log(1-V))$, $1/\theta$ and $1/\theta^2$ in the DP case, respectively. If those moments are finite, by the \gls{CLT},
  $$\frac{T_n-n\mu}{\sqrt n\sigma}\to_d Z,\quad
  \mbox{as }n\to\infty,$$
and, by means of Anscobe's Theorem, one obtains that
  $$\frac{N(t)-t/\mu}{\sqrt{\sigma^2t/\mu^3}}
  \rightarrow_d Z,\quad \mbox{as }t\to\infty$$
It turns out that the normal limit of $N(\log n)$ corresponds to the normal limit of $K_n$,

  $$\frac{K_n-\log n/\mu}{\sqrt{\sigma^2\log n/\mu^{3}}}
  \rightarrow_d Z,\quad \mbox{as }n\to\infty$$

provided that $\E(-\log V)<\infty$. To see why, consider

iid random variables $X_1,X_2,\ldots$ with values in $\N$ such that $\{X_i=k\}=\{U_i\in(1-R_{k-1},1-R_{k})\}$. Hence $\Prob(X_1=k|{\cal R})=\p_k$. We then have that $K_n=\#\{k: X_i=k\mbox{ for at least one $i$ among }1,\ldots,n\}$. Define $M_n=\max\{X_1,\ldots,X_n\}$.

For $U_{1,n}\leq U_{2,n}\leq\ldots \leq U_{n,n}$ denoting the order statistics corresponding to the uniform variates $U_1,\ldots,U_n$, we have
  $M_n
  =\min \{j:\ 1-R_j\geq U_{n,n}\}
  =\min \{j:\ T_j\geq E_{n,n}\}$
upon transformation $x\to -\log(1-x)$, where $E_{n,n}$ is the maximum of an iid sample of size $n$ from the standard exponential distribution. Since
  $N(t)=\max \{n:\ T_n\leq t\}
  =\min \{n:\ T_n\geq t\}-1$
we have
  $M_n-1=N(E_{n,n})$.
\citet{Gne:etal:09} proves the equivalence
  $$\frac{M_n-b_n}{a_n}\to_d X
  \quad\Longleftrightarrow
  \frac{N(\log n)-b_n}{a_n}\to_d X$$
where $X$ is a random variable with a proper and non degenerate distribution with $a_n>0$, $a_n\to\infty$ and $b_n\in\R$. A key fact exploited in the proof is that, from extreme-value theory, $E_{n,n}-\log n$ has a asymptotic distribution of Gumbel type. That $M_n$ can be replaced by $K_n$ in the equivalence relation above follows from the fact that $M_n-K_n$, the number of integers $k<M_n$ not appearing in the sample $X_1,\ldots,X_n$, is bounded in probability when $\E(-\log V)<\infty$, see Proposition 5.1 in \citet{Gne:etal:09}.

Back to the Pitman--Yor process case, by Theorem \ref{th:1} we have
  $n^{-\alpha/(1-\alpha)}
  N(\log n)\to_{d}
  (T_{\alpha,\theta}/\alpha)^{-\alpha/(1-\alpha)}$
while
  $n^{-\alpha} K_n
  \to_{a.s.}

  (T_{\alpha,\theta})^{-\alpha}$,
so we see that $N(\log n)$ and $K_n$ do not have the same asymptotic behavior as in the $\alpha=0$ case.

By using the fact that
  $$\Prob(X_1>n|(\p_i))=R_n,\quad
  R_n\sim_{a.s.}\alpha n^{-(1-\alpha)/\alpha}
  T_{\alpha,\theta}^{-1},$$
and the fact that, conditional on $(\p_i)$, $M_n$ belongs to the domain of attraction of Fr\`echet distribution, \citet[Theorem 6.1]{Pit:Yak:17} establishes that
  $$\Prob(M_n\leq x n^{\alpha/(1-\alpha)})
  \to
  \E\big[\exp\big(-\alpha
  T_{\alpha,\theta}^{-1}
  x^{-(1-\alpha)/\alpha}\big)\big]$$
so we see that $N(\log n)$ and $M_n$ do not have the same asymptotic behavior as in the $\alpha=0$ case, although they share the same growth rate $n^{\alpha/(1-\alpha)}$. Finally, the non correspondence of the asymptotic distribution of $M_n$ and $K_n$ suggests that the behavior of $M_n-K_n$ is radically different with respect to the $\alpha=0$ case.

\section{Random Generation of $T_{\alpha,\theta}$}\label{appendix:devroye}

Let $Y$ be a standard $\mbox{Exp}(1)$ random variable. Note that
\begin{align*}
  \Prob((Y/T_\alpha)^\alpha>x)
  &=\int_0^\infty \Prob(Y>x^{1/\alpha} t)f_\alpha(t)\ddr t
  =\int_0^\infty \exp[-x^{1/\alpha}t]f_\alpha(t)\ddr t\\
  &=\E\edr^{-x^{1/\alpha}T_\alpha}
  =\edr^{-x}=\Prob(Y>x)
\end{align*}
so we have
  $Y=_d(Y/T_\alpha)^\alpha$.
For $r<\alpha$, $\E(Y^{-r/\alpha})<\infty$, so we find that
  $\E(Y^{-r/\alpha})=\E( T_\alpha^r)\E(Y^{-r})$
and
\begin{equation}\label{eq:A1}
  \E( T_\alpha^r)=\frac{\E(Y^{-r/\alpha})}{\E(Y^{-r})}
  =\frac{\Gamma(1-r/\alpha)}{\Gamma(1-r)}.
\end{equation}
The normalizing constant in $f_{\alpha,\theta}(t)$ is
  $\int_0^\infty t^{-\theta}f_\alpha(t)\ddr t
  =\E(T_\alpha^{-\theta})$,
so set $r=-\theta$ and note that $-\theta<\alpha$.
Let $G_a$ is a gamma random variable with shape $a>0$ and unit rate. Simple moment comparisons using \eqref{eq:A1} yield the distributional equality
  $G_{1+\theta/\alpha}\overset{d}{=}
  (G_{1+\theta}/T_{\alpha,\theta})^\alpha$,
which, however, does not provide a way to generate from $T_{\alpha,\theta}$. For this we resort to \citet{Dev:09}. First we recall how to generate a Zoltarev random variable $Z_{\alpha,b}$ for $\alpha\in(0,1)$ and $b=\theta/\alpha>-1$. Let
  $$C=\frac{\Gamma(1+b\alpha)\Gamma(1+b(1-\alpha))}
  {\pi\Gamma(1+b)}$$
and
  $$B(u)=A(u)^{-(1-\alpha)}
  =\frac{\sin(u)}
  {\sin(\alpha u)^\alpha\,
  \sin((1-\alpha)u)^{1-\alpha}}$$
A simple asymptotic argument yields the value
  $B(0)=\alpha^{-\alpha}(1-\alpha)^{-(1-\alpha)}$.
Then
  $f(x)=C\, B(x)^{b},\quad 0\leq x\leq \pi$.
The following bound holds
  $$f(x)\leq
  C B(0)^b\edr^{-\frac{x^2}{2\sigma^2}},\quad
  \text{with }\sigma^2=\frac{1}{b\alpha(1-\alpha)}$$
This Gaussian upper bound suggests a simple rejection sampler for sampling Zolotarev random variates. Following \citet{Dev:09}, it is most efficient to adapt the sampler to the value of $\sigma$. If $\sigma\geq\sqrt{2\pi}$, rejection from a uniform random variate is best. Otherwise, use a normal dominating curve as suggested in the bound above. The details are given below.

\begin{quote}
\centerline{{\sc Algorithm 3} (Sampler of $T_{\alpha,\theta}$)}
\vspace{4pt}

{\tt
1. set $b=\theta/\alpha$ and $\sigma=\sqrt{b\alpha(1-\alpha)}$
\vspace{4pt}

2. if $\sigma\geq\sqrt{2\pi}$:
\\
\phantom{1. }then repeat:$\quad$ generate $U\sim\text{\rm Unif}(0,\pi)$ and $V\sim\text{\rm Unif}(0,1)$.
\\
\phantom{1. then repeat:$\quad$} set $X\leftarrow U$, $W\leftarrow B(X)$.
\\
\phantom{1. then} until $V\leq (W/B(0))^b$

\\
\phantom{1. }else repeat:$\quad$ generate $N\sim\text{\rm N}(0,1)$ and $V\sim\text{\rm Unif}(,1)$.
\\
\phantom{1. else repeat:$\quad$} set $X\leftarrow \sigma|N|$, $W\leftarrow B(X)$.
\\
\phantom{1. else} until $X\leq \pi$ and $V\edr^{-N^2/2}\leq (W/B(0))^b$

\vspace{4pt}

3. generate $G\overset{d}{=}
G_{1+b(1-\alpha)/\alpha}$
\vspace{4pt}

4. set $T\leftarrow 1/(WG^{1-\alpha})^{1/\alpha}$
\vspace{4pt}

5. return $T$
}
\end{quote}

\section*{Acknowledgement}
P. De Blasi and I. Pr\"unster are supported by the European Research Council (ERC), StG \comillas{N-BNP} 306406 and by MIUR, PRIN Project 2015SNS29B.

\bibliographystyle{apalike}
\bibliography{biblio}

\end{document}

config: arxiv


TITLE: Why is force equal to mass into acceleration?
QUESTION [2 upvotes]: If I'm sitting on a chair I'm exerting a force on it but my body isn't even moving so it doesn't even have a velocity. Then how can it even have a change of velocity i.e, acceleration?

REPLY [4 votes]: Force is not equal to mass times acceleration.
Only net force is. This is a very important distinction. It might be helpful to always write Newton's 2nd law as:
$$\sum F=ma$$
rather than just $F=ma$.
When sitting in your chair, as you describe, you are pressed down by your weight. At the same time the chair pushes up with a normal force. Since they balance each other out perfectly, the acceleration is zero, $a=0$. This scenario is basically what we call newton's 1st law.


TITLE: Why Does $\rm{E}[1_A \mid X] = P^X(A\mid X)$?
QUESTION [4 upvotes]: Following the excellent answer here, it is stated that

Connection: Let $P^X(\cdot\mid\cdot)$ be a regular conditional probability of $P$ given $X$.  Then for any $A \in \mathcal{F}$ we have
  $$
{\rm E}[1_A\mid X]=\varphi(X),
$$
  where $\varphi(x) = P^X(A\mid x)$.  In short we write ${\rm E}[1_A\mid X]=P^X(A\mid X)$.

I'm trying to prove this for myself, but I'd appreciate some help.  We need to show two things:

$P^X(A\mid X)$ is $\sigma(X)$-measurable
$\rm{E}[1_C P^X(A\mid X)] = \rm{E}[1_C 1_A]$ for all $C \in \sigma(X)$.

Since $P^X(\cdot\mid\cdot)$ is a regular conditional probability, we know that for fixed $A \in \mathcal{F}$, the mapping $x \mapsto P^X(A\mid x)$ is $(\mathcal{B}(\mathbb{R}),\mathcal{B}(\mathbb{R}))$-measurable.  Hence the composition $P^X(A\mid \cdot) \circ X: \Omega \to \mathbb{R}$ is a random variable for fixed $A \in \mathcal{F}$, which the author writes as $P^X(A\mid \cdot) \circ X = P^X(A|X)$.  Thus $P^X(A\mid X)$ is $\sigma(X)$-measurable, and we have 1.
For 2, recall that for all $A \in \mathcal{F}$ and $B \in \mathcal{B}(\mathbb{R})$ the conditional probability  is defined to satisfy
$$
\int_B P^X(A\mid x) \,d\mu_X(x) = P(A \cap \{X \in B\}),
$$
where $\mu_X$ is the distribution measure of $X$ on $\mathcal{B}(\mathbb{R})$.  This is the part I'm having trouble with.  Here's my attempt:
\begin{align*}
\rm{E}[1_C P^X(A\mid X)] & = \int_{\Omega} 1_C(\omega)P^X(A\mid X)(\omega) dP(\omega) \\
& = \int_{\mathbb{R}} 1_{\{1\}}(x) P^X(A\mid x)d\mu_X(x) &\qquad& (2) \\
& = \int_{\{1\}}P^X(A\mid x)d\mu_X(x) \\
& = P(A \cap \{X = 1\}) \\
& = \int_\Omega 1_A(\omega) 1_{\{X = 1\}}(\omega) dP(\omega) \\
& = \int_\Omega 1_A(\omega) 1_C(\omega) dP(\omega) &\qquad& (6) \\
& = \rm{E}[1_A 1_C].
\end{align*}
I'm almost certain lines (2) and (6) aren't true, but I'm not sure how else to write the indicator random variable when transitioning to the distribution measure in (2).  For (6), I don't see why $\{X = 1\} = C$, either, but it's my best shot at it right now :/

REPLY [2 votes]: Note that $C\in\sigma(X)$ can be written as $X^{-1}(B)=\{X\in B\}$ for some $B\in\mathcal{B}(\mathbb{R})$ and hence $\mathbf{1}_C=\mathbf{1}_B(X)$. Therefore,
$$
\mathrm{E}[\mathbf{1}_CP^X(A\mid X)]=\int_\Omega \mathbf{1}_B(X)P^X(A\mid X)\,\mathrm dP=\int_BP^X(A\mid x)\,P_X(\mathrm dx)=P(A\cap \{X\in B\}).
$$
But
$$
P(A\cap \{X\in B\})=P(A\cap C)=\mathrm{E}[\mathbf{1}_A\mathbf{1}_C]
$$
which is what we want.


text:
\begin{document}

\title[Transport equation with nonlocal velocity]{On the local and global existence of solutions to 1D transport equations with nonlocal velocity}

\author{Hantaek Bae}
\address{Department of Mathematical Sciences, Ulsan National Institute of Science and Technology (UNIST), Republic of Korea}
\email{hantaek@unist.ac.kr}

\author{Rafael Granero-Belinch\'{o}n}
\address{Departamento  de  Matem\'aticas,  Estad\'istica  y  Computaci\'on,  Universidad  de Cantabria,  Avda.  Los  Castros  s/n,  Santander,  Spain.}
\email{rafael.granero@unican.es}

\author{Omar Lazar}
\address{Departamento de An\'alisis Matem\'atico \& IMUS Universidad de Sevilla C/ Tarfia s/n, Campus Reina Mercedes, 41012 Sevilla, Spain}
\email{omarlazar@us.es}

\date{\today}

\keywords{Fluid equations, 1D models, Global weak solution}

\subjclass[2010]{35A01, 35D30, 35Q35, 35Q86}

\begin{abstract}
We consider the 1D transport equation with nonlocal velocity field:
\begin{equation*}\label{intro eq}
\begin{split}
&\theta_t+u\theta_x+\nu \Lambda^{\gamma}\theta=0, \\
& u=\mathcal{N}(\theta),
\end{split}
\end{equation*}
where $\mathcal{N}$ is a nonlocal operator. In this paper, we show the existence of solutions of this model locally and globally in time for various types of nonlocal operators.
\end{abstract}

\maketitle

\vspace{-2em}

\section{Introduction} \label{sec:1}

In this paper, we study transport equations with nonlocal velocity. One of the most well-known equation is  the two dimensional Euler equation in vorticity form,
\[
\omega_{t}+u\cdot \nabla \omega=0,
\]
where the velocity $u$ is recovered from the vorticity $\omega$ through
\[
u=\nabla^\perp(-\Delta)^{-1}\omega \quad \text{or equivalently} \quad  \widehat{u}(\xi)=\frac{i\xi^{\perp}}{|\xi|^{2}}\widehat{\omega}(\xi).
\]
Other nonlocal and quadratically nonlinear equations, such as the surface quasi-geostrophic equation, the incompressible porous medium equation, Stokes equations, magneto-geostrophic equation in multi-dimensions, have  been studied intensively as one can see in  \cite{Bae 2, Bae, Baker, Carrillo, CC, CC2, Chae, De Gregorio, Kiselev, Lazar2, LiRodrigo, LiRodrigo2, Morlet} and references therein.

We here consider the 1D transport equations with nonlocal velocity field of the form
\begin{subequations}\label{model equation}
\begin{align}
&\theta_t+u\theta_x+\nu \Lambda^{\gamma}\theta=0,\\
& u=\mathcal{N}(\theta),
\end{align}
\end{subequations}
where $\mathcal{N}$ is typically expressed by a Fourier multiplier. The study of (\ref{model equation}) is mainly motivated by \cite{CCF} where C\'ordoba, C\'ordoba, and Fontelos proposed the following 1D model
\begin{subequations}\label{CCF}
\begin{align}
&\theta_{t}+u\theta_{x}=0, \\
& u=-\mathcal{H}\theta, \quad (\text{$\mathcal{H}$ being the Hilbert transform})
\end{align}
\end{subequations}
for the 2D surface quasi-geostrophic equation and  proved the finite time blow-up of smooth solutions. In this paper, we deal with (\ref{CCF}) and its variations with the following objectives.
\begin{enumerate}[]
\item (1) The existence of weak solution with \emph{rough initial data}. The existence of global-in-time solutions is possible even if strong solutions blow up in finite time, as in the case of the Burgers' equation.
\item (2) The existence of strong solution when the velocity $u$ is more singular than $\theta$. We intend to see  the competitive relationship between nonlinear terms and viscous terms.
\end{enumerate}

More specifically, the topics covered in this paper can be summarized as follows.

\vspace{1ex}

\noindent
\textbullet \ {\bf The model 1: $\mathcal{N}=-\mathcal{H}$ and $\nu=0$.} We first show the existence of local-in-time solution in a critical space under the scaling $\theta_{0}(x)\mapsto \theta_{0}(\lambda x)$.  We then introduce the notion of a weak super-solution and obtain a global-in-time weak super-solution with $\theta_{0}\in L^{1}\cap L^{\infty}$ and $\theta_{0}\ge 0$.

\vspace{1ex}

\noindent
\textbullet \ {\bf The model 2: $\mathcal{N}=-\mathcal{H}(\partial_{xx} )^{-\alpha}$, $\alpha>0$, $\nu=1$, and $\gamma>0$.} This is a regularized version of (\ref{CCF}) which is also closely related to many equations as mentioned in \cite{Bae 3}. In this case, we show the existence of weak solutions globally in time under weaker conditions on $\alpha$ and $\gamma$ compared to \cite{Bae 3}.

\vspace{1ex}

\noindent
\textbullet \ {\bf The model 3: $\mathcal{N}=-\mathcal{H}(\partial_{xx} )^{\beta}$, $\beta>0$, $\nu=1$, and $\gamma>0$.} Since $\beta>0$, the velocity field is more singular than the previous two models.  In this case, we show the existence of strong solutions locally in time in two cases: (1) $0<\beta\leq \frac{\gamma}{4}$ when $0<\gamma<2$ and (2) $0<\beta<1$ when $\gamma=2$. We also show the existence of strong solutions for $0<\beta<\frac{1}{2}$ and $\gamma=2$ with rough initial data. We finally show the existence of strong solutions globally in time with $0<\beta<\frac{1}{4}$ and $\gamma=2$.

\vspace{1ex}

We will give detailed statements and proofs of our results in Section 3--5.

\section{Preliminaries}

All constants will be denoted by $C$ that is a generic constant. In a series of inequalities, the value of $C$ can vary with each inequality. We use following notation: for a Banach space $X$,
\[
C_{T}X=C([0,T]:X), \quad L^{p}_{T}X=L^{p}(0,T:X).
\]

The Hilbert transform is defined as
\[
\mathcal{H}f(x)=\text{p.v.} \int_{\mathbb{R}} \frac{f(y)}{x-y}dy.
\]
We will use the BMO space (see e.g. \cite{Bahouri} for the definition) and its dual which is the Hardy space $\mathcal{H}^1$ which consists of those $f$ such that $f$ and $\mathcal{H}f$ are integrable. We will use the following formula
\[
2 \mathcal{H}(f\mathcal{H}f)=(\mathcal{H}f)^2 - f^2
\]
which implies that $g=f\mathcal{H}f \in \mathcal{H}^1$ and for any $f\in L^{2}$,
\begin{equation} \label{hardy}
\Vert g \Vert_{\mathcal{H}^1}  \leq \Vert f \Vert^{2}_{L^{2}}.
\end{equation}

The differential operator $\Lambda^{\gamma}=(\sqrt{-\Delta})^{\gamma}$ is defined by the action of the following kernels \cite{Cordoba}:
\eqn \label{lambda gamma}
\Lambda^{\gamma} f(x)=c_{\gamma}\text{p.v.} \int_{\mathbb{R}} \frac{f(x)-f(y)}{|x-y|^{1+\gamma}}dy,
\een
where $c_{\gamma}>0$ is a normalized constant. Alternatively, we can define $\Lambda^{\gamma}=(\sqrt{-\Delta})^{\gamma}$ as a Fourier multiplier: $\widehat{\Lambda^{\gamma} f}(\xi)=|\xi|^{\gamma}\widehat{f}(\xi)$. When $\gamma=1$,  $\Lambda f(x)=\mathcal{H}f_{x}(x)$.

\vspace{1ex}

We finally introduce Simon's compactness.

\begin{lemma} \cite{Simon} \label{lem:2.2}
Let $X_{0}$, $X_{1}$, and $X_{2}$ be Banach spaces such that $X_{0}$ is compactly embedded in $X_{1}$ and $X_{1}$ is a subset of $X_{2}$. Then, for $1\leq p<\infty$, the set $\left\{v\in L^{p}_{T}X_{0}: \ \frac{\partial v}{\partial t}\in L^{1}_{T}X_{2}\right\}$ is compactly embedded in $L^{p}_{T}X_{1}$.
\end{lemma}

\section{The model 1}

We now study (\ref{model equation}) with $\mathcal{N}=-\mathcal{H}$ and $\nu=0$ which is nothing but (\ref{CCF}):
\begin{subequations} \label{eq:1.1}
\begin{align}
& \theta_{t} -\left(\mathcal{H}\theta\right)\theta_{x} =0,  \label{eq:1.1 a}\\
& \theta(0,x)=\theta_{0}(x). \label{eq:1.1 b}
\end{align}
\end{subequations}

\subsection{Local well-posedness}

The local well-posedness of (\ref{eq:1.1}) is established in $H^{2}$ (\cite{Bae}) and  $H^{\frac{3}{2}-\gamma}$ with the viscous term $\Lambda^{\gamma}\theta$ (\cite{HDong}). To improve these results, we first notice that (\ref{eq:1.1}) has the following scaling invariant property: if $\theta(t,x)$ is a solution of (\ref{eq:1.1}), then so is $\theta_{\lambda}(t,x)=\theta(\lambda t, \lambda x)$. So, we take initial data in a space whose norm is closely invariant under the scaling: $\theta_{0}(x)\mapsto \theta_{\lambda 0}(x)=\theta_{0}(\lambda x)$. In this paper, we take the space $\dot{B}^{\frac{3}{2}}_{2,1}$ because there is a constant $C$ such that
\[
C^{-1} \left\|\theta_{\lambda 0}\right\|_{\dot{B}^{\frac{3}{2}}_{2,1}} \leq \|\theta_{0}\|_{\dot{B}^{\frac{3}{2}}_{2,1}} \leq C \left\|\theta_{\lambda 0}\right\|_{\dot{B}^{\frac{3}{2}}_{2,1}}.
\]
The mathematical tools needed to prove the local well-posedness of (\ref{eq:1.1}), such as the Littlewood-Paley decomposition and Besov spaces, are provided in the appendix. We also need the following commutator estimate \cite[Lemma 2.100, Remark 2.101]{Bahouri}.

\begin{lemma}[Commutator estimate]\label{commutator lemma 1}
For $f, g \in \mathcal{S}$
\[
\left\|[f,\Delta_{j}]g_{x}\right\|_{L^{2}}\leq Cc_{j}2^{-\frac{3}{2}j} \left\|f_{x}\right\|_{\dot{B}^{\frac{1}{2}}_{2,1}} \left\|g\right\|_{\dot{B}^{\frac{3}{2}}_{2,1}}, \quad \sum^{\infty}_{j=-\infty}c_{j}\leq 1.
\]
\end{lemma}

The first result in this paper the following theorem.

\begin{theorem}\label{Besov theorem}
For any $\theta_{0}\in \dot{B}^{\frac{3}{2}}_{2,1}$, there exists $T=T(\|\theta_0\|)$ such that a unique solution of (\ref{eq:1.1}) exists in  $C_{T}\dot{B}^{\frac{3}{2}}_{2,1}$.
\end{theorem}

\begin{proof}
We only provide a priori estimates of $\theta$ in the space stated in Theorem \ref{Besov theorem}.  The other parts, including the approximation procedure, are rather standard.

We apply $\Delta_{j}$ to (\ref{eq:1.1}), multiply by  $\Delta_{j}\theta$, and integrate the resulting equation over $\mathbb{R}$ to get
\begin{equation} \label{eq:3.2}
\begin{split}
\frac{1}{2}\frac{d}{dt} \left\|\Delta_{j} \theta \right\|^2_{L ^{2}} &=\int_{\mathbb{R}} \Delta_{j}\left((\mathcal{H}\theta)\theta_{x} \right) \Delta_{j}\theta dx\\
&=\int_{\mathbb{R}} \left((\mathcal{H}\theta)\Delta_{j}\theta_{x} \right) \Delta_{j}\theta dx+ \int_{\mathbb{R}} \left[\Delta_{j}, \mathcal{H}\theta\right]\Delta_{j}\theta_{x} \Delta_{j}\theta dx\\
&=-\frac{1}{2}\int_{\mathbb{R}} (\mathcal{H}\theta)_{x}\left|\Delta_{j}\theta\right|^{2}  dx+ \int_{\mathbb{R}} \left[\Delta_{j}, \mathcal{H}\theta\right]\Delta_{j}\theta_{x} \Delta_{j}\theta dx.
\end{split}
\end{equation}
By the Bernstein inequality, we have
\eqn \label{low frequency}
\left\|\mathcal{H}\theta_{x}\right\|_{L^{\infty}}\leq C\|\theta\|_{\dot{B}^{\frac{3}{2}}_{2,1}}.
\een
We then apply Lemma \ref{commutator lemma 1} to the second term in the right-hand side of (\ref{eq:3.2}) to obtain
\begin{equation}\label{eq:3.3}
\begin{split}
\int_{\mathbb{R}} \left[\Delta_{j}, \mathcal{H}\theta\right]\Delta_{j}\theta_{x} \Delta_{j}\theta dx\leq Cc_{j}2^{-\frac{3}{2}j}\|\theta\|^{2}_{\dot{B}^{\frac{3}{2}}_{2,1}} \left\|\Delta_{j} \theta \right\|_{L ^{2}}.
\end{split}
\end{equation}
By (\ref{eq:3.2}), (\ref{low frequency}), and (\ref{eq:3.3}), we have
\[
\frac{d}{dt}\|\theta\|^{2}_{\dot{B}^{\frac{3}{2}}_{2,1}}\leq C\|\theta\|^{3}_{\dot{B}^{\frac{3}{2}}_{2,1}},
\]
from which we deduce
\[
\|\theta(t)\|_{\dot{B}^{\frac{3}{2}}_{2,1}}\leq \frac{\|\theta_{0}\|_{\dot{B}^{\frac{3}{2}}_{2,1}}}{1-Ct\|\theta_{0}\|_{\dot{B}^{\frac{3}{2}}_{2,1}}} \leq 2 \|\theta_{0}\|_{\dot{B}^{\frac{3}{2}}_{2,1}} \quad \text{for all $t\leq T=\frac{1}{2C \|\theta_{0}\|_{\dot{B}^{\frac{3}{2}}_{2,1}}}$}.
\]
This completes the proof.
\end{proof}

\subsection{Global weak super-solution}

We next consider (\ref{eq:1.1}) with rough initial data. More precisely, we assume that $\theta_{0}$ satisfies the following conditions
\eqn \label{initial condition}
\theta_{0}\ge 0, \quad \theta_{0}\in L^{1}\cap L^{\infty}.
\een
Since $\theta$ satisfies the transport equation, we have
\eqn
\theta(t,x)\ge 0, \quad \theta \in L^{\infty}(\mathbb{R}) \quad \text{for all time}.
\een

If we follow the usual weak formulation of (\ref{eq:1.1}), for all $\phi\in C^{\infty}_{c}([0,\infty)\times \mathbb{R})$
\eqn \label{weak solution to 1D sqg}
\int^{T}_{0}\int_{\mathbb{R}} \left[\theta \psi_{t} - \left(\mathcal{H}\theta\right)\theta \psi_{x}+\left(\Lambda \theta\right) \theta \psi \right]  dxdt =\int_{\mathbb{R}}\theta_{0}(x)\psi(x,0)dx.
\een
For $\theta_{0}\ge 0$, there is gain of a half derivative from the structure of the nonlinearity, that is
\eqn \label{half energy dd}
\left\|\theta(t)\right\|_{L^{1}} +\int^{t}_{0}\left\|\Lambda^{\frac{1}{2}}\theta(s)\right\|^{2}_{L^{2}}ds=\left\|\theta_{0}\right\|_{L^{1}}.
\een
So, we can rewrite the left-hand side of (\ref{weak solution to 1D sqg}) as
\[
\int^{T}_{0}\int_{\mathbb{R}} \left[\theta \psi_{t} - \left(\mathcal{H}\theta\right)\theta \psi_{x}+\Lambda^{\frac{1}{2}}\theta \left[\Lambda^{\frac{1}{2}},\psi\right]\theta +\left|\Lambda^{\frac{1}{2}}\theta\right|^{2}\psi\right]  dxdt =\int_{\mathbb{R}}\theta_{0}(x)\psi(x,0)dx.
\]
However,  the $\dot{H}^{\frac{1}{2}}$ regularity derived from (\ref{half energy dd}) is not enough to pass to the limit in
\[
\int^{T}_{0}\int_{\mathbb{R}} \left|\Lambda^{\frac{1}{2}}\theta^{\epsilon}\right|^{2}\psi dxdt
\]
from the $\epsilon$-regularized equations  described below. So, we introduce a new notion of solution. Let
\[
\mathcal{A}_{T}=L^{\infty}_{T}\left(L^{1}\cap L^{\infty}\right)\cap L^{2}_{T}H^{\frac{1}{2}}.
\]

\begin{definition}
We say $\theta$ is a weak super-solution of (\ref{eq:1.1}) on the time interval $[0,T]$ if $\theta(t,x)\ge 0$ for all $t\in [0,T]$, $\theta\in \mathcal{A}_{T}$, and for each nonnegative $\psi\in C^{\infty}_{c}([0,T]\times\mathbb{R})$,
\eqn \label{supersolution to 1d sqg}
\int^{T}_{0}\int_{\mathbb{R}} \left[\theta \psi_{t} - \left(\mathcal{H}\theta\right)\theta \psi_{x}+\Lambda^{\frac{1}{2}}\theta \left[\Lambda^{\frac{1}{2}},\psi\right]\theta +\left|\Lambda^{\frac{1}{2}}\theta\right|^{2}\psi\right]  dxdt \ge \int_{\mathbb{R}}\theta_{0}(x)\psi(x,0)dx.
\een
\end{definition}

To prove Theorem \ref{main theorem}, we need to estimate a commutator term involving $\Lambda^{\frac{1}{2}}$:
\[
\left[\Lambda^{1/2},\psi\right](f-g)\in L^{6}
\]
which is proved in \cite{Bae 3}.

\begin{lemma} \label{commutator lemma 0}
For $f\in L^{\frac{3}{2}}$, $g\in L^{\frac{3}{2}}$  and $\psi \in W^{1,\infty}$, we have
\[
\left\| \left[\Lambda^{\frac{1}{2}},\psi \right]f- \left[\Lambda^{\frac{1}{2}},\psi \right]g\right\|_{L^{6}}\leq C\|\psi\|_{W^{1,\infty}} \left\|f-g\right\|_{L^{\frac{3}{2}}}.
\]
\end{lemma}

The second result in our paper is the following theorem.

\begin{theorem}\label{main theorem}
For any  $\theta_{0}$ satisfying (\ref{initial condition}), there exists a weak super-solution of (\ref{eq:1.1}) in $\mathcal{A}_{T}$.
\end{theorem}

\begin{proof}
We first regularize initial data as $\theta^{\epsilon}_{0}=\rho_{\epsilon}\ast \theta_{0}$ where $\rho_{\epsilon}$ is a standard mollifier that preserve the positivity of the regularized initial data. We then regularize the equation by introducing the Laplacian term with a coefficient $\epsilon>0$, namely
\eqn \label{regularized equation}
\theta^{\epsilon}_t -\mathcal{H}\theta^{\epsilon}\theta^{\epsilon}_x =\epsilon \theta^{\epsilon}_{xx}.
\een
For the proof of the existence of a global-in-time smooth solution we refer to \cite{Lazar}. Moreover, $\theta^{\epsilon}$ satisfies that $\theta^{\epsilon}\ge 0 $ and
\[
\left\| \theta^{\epsilon}(t) \right\|_{L^{1}}+\left\| \theta^{\epsilon}(t) \right\|_{L^{\infty}}+ \int^{t}_{0}\left\|\Lambda^{\frac{1}{2}}\theta^{\epsilon}(s)\right\|^{2}_{L^{2}}ds \leq  \left\|\theta_{0}\right\|_{L^{1}} + \left\|\theta_{0}\right\|_{L^{\infty}}.
\]
Therefore, $(\theta_{\epsilon})$ is bounded in $\mathcal{A}_{T}$ uniformly in $\epsilon>0$.

From this, we have uniform bounds
\[
\mathcal{H}\theta^{\epsilon}\in L^{4}_{T}L^{2}, \quad \theta^{\epsilon}\in L^{2}_{T}L^{2}, \quad \left(\left(\mathcal{H}\theta^{\epsilon}\right)\theta^{\epsilon}\right)_x\in L^{\frac{4}{3}}_{T}H^{-2}, \quad \epsilon \theta^{\epsilon}_{xx}\in L^{2}_{T}H^{-2}.
\]
Moreover,  for any $\phi \in H^{2}$,
\[
\int_{\mathbb{R}}\left|\theta^{\epsilon} \Lambda\theta^{\epsilon} \phi \right|  dx \leq \left\|\Lambda^{\frac{1}{2}}\theta^{\epsilon}\right\|^{2}_{L^{2}} \left\|\phi\right\|_{L^{\infty}} + \left\|\Lambda^{\frac{1}{2}}\theta^{\epsilon}\right\|_{L^{2}} \left\|\theta^{\epsilon}\right\|_{L^{\infty}} \left\|\Lambda^{\frac{1}{2}}\phi\right\|_{L^{\infty}}
\]
which implies that
\[
\theta^{\epsilon} \Lambda\theta^{\epsilon} \in L^{1}_{T}H^{-2}.
\]

Combining all together, we obtain
\[
\theta^{\epsilon}_{t}=\mathcal{H}\theta^{\epsilon}\theta^{\epsilon}_x +\epsilon \theta^{\epsilon}_{xx}= \left(\mathcal{H}\theta^{\epsilon}\theta^{\epsilon}\right)_x -\theta^{\epsilon} \Lambda\theta^{\epsilon}+\epsilon \theta^{\epsilon}_{xx}\in L^{1}_{T}H^{-2}.
\]

To pass to the limit into the weak super-solution formulation, we extract a subsequence of $\left(\theta^{\epsilon}\right)$, using the same index $\epsilon$ for simplicity, and a function $\theta \in \mathcal{A}_T$ such that
\begin{equation}\label{convergence of approximate solutions}
\begin{split}
& \theta^{\epsilon} \stackrel{\star}{\rightharpoonup} \theta \quad \text{in} \quad L^{\infty}_{T}\left(L^{p}\cap H^{\frac{1}{2}}\right) \quad \text{for all $p\in (1,\infty)$},\\
& \theta^{\epsilon} \rightharpoonup \theta \quad \text{in} \quad L^{2}_{T}H^{\frac{1}{2}},\\
& \theta^{\epsilon} \rightarrow  \theta \quad \text{in $L^{2}_{T}L^{p}$ for all $1<p<\infty$} ,
\end{split}
\end{equation}
where we use Lemma \ref{lem:2.2} for the strong convergence with
\[
X_0=L^2_{T}H^{\frac{1}{2}},\quad X_{1}=L^2_{T}L^p,\quad X_2=L^1_{T}H^{-2}.
\]

We now multiply (\ref{regularized equation}) by a test function $\psi\in \mathcal{C}^{\infty}_{c}\left([0,T)\times\mathbb{R}\right)$ and integrate over $\mathbb{R}$. Then,
\begin{equation} \label{weak form with commutator 111}
\begin{split}
&\int^{T}_{0}\int \Big[\theta^{\epsilon} \psi_{t} -  \underbrace{\left(\mathcal{H}\theta^{\epsilon}\right)\theta^{\epsilon}\psi_{x}}_{\text{I}}+ \epsilon \theta^{\epsilon}\psi_{xx}\Big]  dxdt - \int \theta^{\epsilon}_{0}(x)\psi(0,x)dx\\
& = -\int^{T}_{0}\int \underbrace{\Lambda^{\frac{1}{2}}\theta^{\epsilon} \left[\Lambda^{\frac{1}{2}},\psi \right]\theta^{\epsilon}}_{\text{II}}  dxdt -\int^{T}_{0}\int \underbrace{\left|\Lambda^{\frac{1}{2}}\theta^{\epsilon}\right|^{2}\psi}_{\text{III}}  dxdt.
\end{split}
\end{equation}

We note that we are able to rearrange terms in the usual weak formulation into (\ref{weak form with commutator 111}) since $\theta^{\epsilon}$ is smooth.  By the strong convergence in (\ref{convergence of approximate solutions}), we can pass to the limit to $\text{I}$. Moreover, since
\[
\left[\Lambda^{\frac{1}{2}},\psi \right]\theta^{\epsilon} \rightarrow \left[\Lambda^{\frac{1}{2}},\psi \right]\theta
\]
strongly in $L^{2}_{T}L^{6}$ by Lemma \ref{commutator lemma 0} and the strong convergence in (\ref{convergence of approximate solutions}), we can pass to the limit to $\text{II}$. Lastly, by Fatou's lemma,
\[
\lim_{\epsilon\rightarrow 0}\int^{T}_{0}\int \left|\Lambda^{\frac{1}{2}}\theta^{\epsilon}\right|^{2}\psi dxdt \ge \int^{T}_{0}\int \left|\Lambda^{\frac{1}{2}}\theta\right|^{2}\psi dxdt.
\]
Combining all the limits together, we obtain that
\begin{equation} \label{weak form with commutator 2}
\begin{split}
\int^{T}_{0}\int_{\mathbb{R}} \left[\theta \psi_{t} - \left(\mathcal{H}\theta\right)\theta \psi_{x}+\Lambda^{\frac{1}{2}}\theta \left[\Lambda^{\frac{1}{2}},\psi\right]\theta +\left|\Lambda^{\frac{1}{2}}\theta\right|^{2}\psi\right]  dxdt \ge \int_{\mathbb{R}}\theta_{0}(x)\psi(x,0)dx.
\end{split}
\end{equation}
This completes the proof.
\end{proof}

\section{The model 2}

We now consider the following equation:
\begin{subequations} \label{new model}
\begin{align}
& \theta_{t} -\left(\mathcal{H}(\partial_{xx})^{-\alpha}\theta\right)\theta_{x} +\Lambda^{\gamma}\theta=0, \\
& \theta(0,x)=\theta_{0}(x),
\end{align}
\end{subequations}
where $\alpha,\gamma>0$. In this case, we focus on the existence of weak solutions under some conditions  of $(\alpha,\gamma)$. As before, we assume that $\theta_{0}$ satisfies the following conditions
\eqn \label{initial condition 2}
\theta_{0}\ge 0, \quad \theta_{0}\in L^{1}\cap L^{\infty}.
\een

Let
\[
\mathcal{B}_{T}=L^{\infty}_{T}\left(L^{1}\cap L^{\infty}\right)\cap L^{2}_{T}H^{\frac{\gamma}{2}}.
\]

\begin{definition}
We say $\theta$ is a weak solution of (\ref{new model}) on the time interval $[0,T]$ if $\theta(t,x)\ge 0$ for all $t\in [0,T]$, $\theta\in \mathcal{B}_{T}$, and for each $\psi\in C^{\infty}_{c}([0,T]\times\mathbb{R})$,
\[
\int^{T}_{0}\int_{\mathbb{R}} \left[\theta \psi_{t} - \left(\mathcal{H}(\partial_{xx})^{-\alpha}\theta\right)\theta \psi_{x}-\Lambda^{1-\frac{\gamma}{2}} (\partial_{xx})^{-\alpha}\theta \Lambda^{\frac{\gamma}{2}} (\theta\psi) -\theta \Lambda^{\gamma}\psi\right]  dxdt = \int_{\mathbb{R}}\theta_{0}(x)\psi(x,0)dx.
\]
\end{definition}

The third result in the paper is the following.

\begin{theorem}\label{new model weak}
Suppose that two positive numbers $\alpha$ and $\gamma$ satisfy
\eqn \label{range of parameters}
0<\gamma<1, \quad \alpha\ge \frac{1}{2}-\frac{\gamma}{2}.
\een
Then, for any  $\theta_{0}$ satisfying (\ref{initial condition 2}), there exists a weak solution of (\ref{new model}) in $\mathcal{B}_{T}$ for all $T>0$.
\end{theorem}

\begin{proof}
As in the proof of Theorem \ref{main theorem}, we regularize $\theta_{0}$ and the equation as
\eqn \label{regularized equation new model}
\theta^{\epsilon}_{0}=\rho_{\epsilon}\ast \theta_{0}, \quad \theta^{\epsilon}_t -\left(\mathcal{H}(\partial_{xx})^{-\alpha}\theta^{\epsilon}\right)\theta^{\epsilon}_{x} +\Lambda^{\gamma}\theta^{\epsilon} =\epsilon \theta^{\epsilon}_{xx}.
\een
Then, the corresponding $\theta^{\epsilon}$ satisfies
\eqn \label{sign and max}
\theta^{\epsilon}(t,x)\ge 0, \quad \left\|\theta^{\epsilon}(t)\right\|_{L^{\infty}}\leq  \|\theta_{0}\|_{L^{\infty}} \quad \text{for all time }
\een
and
\eqn \label{L1 bound new}
\left\|\theta^{\epsilon}(t)\right\|_{L^{1}} +\int^{t}_{0}\left\|\Lambda^{\frac{1}{2}}(\partial_{xx})^{-\frac{\alpha}{2}}\theta^{\epsilon}(s)\right\|^{2}_{L^{2}}ds\leq \left\|\theta_{0}\right\|_{L^{1}}.
\een
We next multiply (\ref{regularized equation new model}) by $\theta^{\epsilon}$ and integrate over $\mathbb{R}$. Then,
\begin{equation*}
\begin{split}
&\frac{1}{2}\frac{d}{dt}\left\|\theta^{\epsilon}(t)\right\|^{2}_{L^{2}}+\left\|\Lambda^{\frac{\gamma}{2}}\theta^{\epsilon}(t)\right\|^{2}_{L^{2}} +\epsilon \left\|\theta^{\epsilon}_{x}\right\|^{2}_{L^{2}}=-\frac{1}{2}\int_{\mathbb{R}}\left\{\Lambda (\partial_{xx})^{-\alpha}\theta^{\epsilon}(t)\right\} (\theta^{\epsilon}(t))^{2}dx\\
&=-\frac{1}{2}\int_{\mathbb{R}}\left\{\Lambda^{1-\frac{\gamma}{2}} (\partial_{xx})^{-\alpha}\theta^{\epsilon}(t)\right\} \Lambda^{\frac{\gamma}{2}}(\theta^{\epsilon}(t))^{2}dx\\
& \leq C\left\|\Lambda^{1-\frac{\gamma}{2}} (\partial_{xx})^{-\alpha}\theta^{\epsilon}(t)\right\|_{L^{2}} \left\|\Lambda^{\frac{\gamma}{2}}\theta^{\epsilon}(t)\right\|_{L^{2}}\|\theta^{\epsilon}(t)\|_{L^{\infty}} \\
& \leq \frac{1}{2} \left\|\Lambda^{\frac{\gamma}{2}}\theta^{\epsilon}(t)\right\|^{2}_{L^{2}} + C\left\|\Lambda^{1-\frac{\gamma}{2}} (\partial_{xx})^{-\alpha}\theta^{\epsilon}(t)\right\|^{2}_{L^{2}}\|\theta^{\epsilon}(t)\|^{2}_{L^{\infty}}.
\end{split}
\end{equation*}
By (\ref{range of parameters}), (\ref{sign and max}) and (\ref{L1 bound new}), we obtain
\eqn \label{new L2 bound}
\|\theta^{\epsilon}(t)\|^{2}_{L^{2}}+\int^{t}_{0}\left\|\Lambda^{\frac{\gamma}{2}}\theta^{\epsilon}(s)\right\|^{2}_{L^{2}}ds +\epsilon \int^{t}_{0}\left\|\theta^{\epsilon}_{x}(s)\right\|^{2}_{L^{2}}ds \leq C\|\theta_{0}\|^{2}_{L^{1}} \|\theta_{0}\|^{2}_{L^{\infty}}.
\een
Therefore, $(\theta_{\epsilon})$ is bounded in $\mathcal{B}_{T}$ uniformly in $\epsilon>0$.

From this, we have uniform bounds
\[
\left\{\left(\mathcal{H}(\partial_{xx})^{-\alpha}\theta\right)\theta\right\}_{x} \in L^{2}_{T}L^{2}, \quad \Lambda^{\gamma}\theta^{\epsilon}+\epsilon \theta^{\epsilon}_{xx}\in L^{2}_{T}H^{-2}.
\]
Moreover, the condition (\ref{range of parameters}) implies that
\[
\left(\Lambda(\partial_{xx})^{-\alpha}\theta^{\epsilon}\right)\theta^{\epsilon}  \in L^{1}_{T}H^{-1}.
\]

Combining all together, we also derive that
\[
\theta^{\epsilon}_{t}\in L^{1}_{T}H^{-2}.
\]

We now multiply (\ref{regularized equation new model}) by a test function $\psi\in \mathcal{C}^{\infty}_{c}\left([0,T)\times\mathbb{R}\right)$ and integrate over $\mathbb{R}$. Then,
\begin{equation} \label{weak form with commutator}
\begin{split}
&\int^{T}_{0}\int \Big[\theta^{\epsilon} \psi_{t} -  \underbrace{\left(\mathcal{H}(\partial_{xx})^{-\alpha}\theta^{\epsilon}\right)\theta^{\epsilon}\psi_{x}}_{\text{I}}+ \Lambda^{\gamma}\theta^{\epsilon}+\epsilon \theta^{\epsilon}\psi_{xx}\Big]  dxdt - \int \theta^{\epsilon}_{0}(x)\psi(0,x)dx\\
& = \int^{T}_{0}\int \underbrace{\Lambda^{1-\frac{\gamma}{2}}\mathcal{H}(\partial_{xx})^{-\alpha}\theta^{\epsilon} \Lambda^{\frac{\gamma}{2}}(\theta^{\epsilon}\psi)}_{\text{II}}  dxdt.
\end{split}
\end{equation}

To pass the limit to this formulation, we extract a subsequence of $\left(\theta^{\epsilon}\right)$, using the same index $\epsilon$ for simplicity, and a function $\theta \in \mathcal{B}_T$ such that
\begin{equation}\label{convergence of approximate solutions 2}
\begin{split}
& \theta^{\epsilon} \stackrel{\star}{\rightharpoonup} \theta \quad \text{in} \quad L^{\infty}_{T}\left(L^{p}\cap H^{\frac{1}{2}}\right) \quad \text{for all $p\in (1,\infty)$},\\
& \theta^{\epsilon} \rightharpoonup \theta \quad \text{in} \quad L^{2}_{T}H^{\frac{\gamma}{2}},\\
& \theta^{\epsilon} \rightarrow  \theta \quad \text{in $L^{2}_{T}H^{1-\frac{\gamma}{2}-2\alpha}\cap L^{2}_{T}L^{p}$ for all $1<p<\infty$} ,
\end{split}
\end{equation}
where we use Lemma \ref{lem:2.2} for the strong convergence with the condition (\ref{range of parameters}) and
\[
X_0=L^2_{T}H^{\frac{\gamma}{2}},\quad X_{1}=L^{2}_{T}H^{1-\frac{\gamma}{2}-2\alpha} \cap L^2_{T}L^p,\quad X_2=L^1_{T}H^{-2}.
\]
By the strong convergence in (\ref{convergence of approximate solutions 2}), we can pass to the limit to $\text{I}$ and $\text{II}$ in (\ref{weak form with commutator}). Therefore, we obtain
\[\int^{T}_{0}\int_{\mathbb{R}} \left[\theta \psi_{t} - \left(\mathcal{H}(\partial_{xx})^{-\alpha}\theta\right)\theta \psi_{x}-\Lambda^{1-\frac{\gamma}{2}} (\partial_{xx})^{-\alpha}\theta \Lambda^{\frac{\gamma}{2}} (\theta\psi) -\theta \Lambda^{\gamma}\psi\right]  dxdt = \int_{\mathbb{R}}\theta_{0}(x)\psi(x,0)dx.
\]
This completes the proof of Theorem \ref{new model weak}.
\end{proof}

\noindent
{\bf Remark.} Theorem \ref{new model weak} improves Theorem 1.4 in \cite{Bae 3}, where $(\alpha, \gamma)$ is assumed to satisfy $\alpha\ge \frac{1}{2}-\frac{\gamma}{4}$. The main idea of taking weaker  regularization in (\ref{new model}) is that the Hilbert transform in front of $(1-\partial_{xx})^{-\alpha}$ gives (\ref{L1 bound new}) which makes to obtain (\ref{new L2 bound}). We choose $\alpha> \frac{1}{2}-\frac{\gamma}{2}$ instead of $ \alpha\ge \frac{1}{2}-\frac{\gamma}{2}$ to apply  compactness argument when we pass to the limit to $\epsilon$-regularized equations.

\section{The model 3}

In this section, we consider the following equation
\begin{subequations} \label{singular model d}
\begin{align}
& \theta_{t} -\left(\mathcal{H}(\partial_{xx})^{\beta}\theta\right)\theta_{x} +\Lambda^{\gamma}\theta=0, \\
& \theta(0,x)=\theta_{0}(x)
\end{align}
\end{subequations}
where $\beta,\gamma>0$. Depending on the range of $\beta$ and $\gamma$, we will have four different results.

\subsection{Local well-posedness}

We begin with the local well-posedness result.

\begin{theorem}  \label{LW and blow-up criterion theorem}
Let $0<\gamma<2$ and $0<\beta\leq \frac{\gamma}{4}$. For $\theta_0 \in H^2 (\Bbb R)$ there exists $T=T(\|\theta_0\|_{H^2})$ such that a unique solution of (\ref{singular model d}) exists in $C\left([0, T); H^2 (\Bbb R)\right)$. Moreover, we have the following blow-up criterion:
\eqn \label{blow-up criterion 1}
\lim\sup_{t\nearrow T^*} \|\theta(t)\|_{H^2}=\infty \quad \text{if and only if} \quad \int_0 ^{T^*} \Big(\left\|u_x(s)\right\|_{L^{\infty}}+\left\|\theta_x(s)\right\|_{L^{\infty}}\Big)ds=\infty.
\een
\end{theorem}

\begin{proof}
Let $u=-\mathcal{H}(\partial_{xx})^{\beta}\theta$. Operating $\partial_x^l$ on (\ref{singular model d}), taking its $L^2$ inner product with $\partial_x^l \theta$, and summing over $l=0, 1,2$,
\begin{equation} \label{pr1}
\begin{split}
&\frac12 \frac{d}{dt} \|\theta(t)\|_{H^2}^2 + \left\|\Lambda^{\frac{\gamma}{2}}\theta\right\|_{H^2}^2= -\sum_{l=0} ^2 \int \partial_x^l (u \theta_x) \partial_x^l \theta dx \\
 &=-\sum_{l=0} ^2 \int \left( \partial_x^l (u \theta_x)- u  \partial_x^l \theta_x \right) \partial_x^l \theta dx-\sum_{l=0} ^2 \int u  \partial_x^l \theta_x  \partial_x^l \theta    dx = \text{I}_1+ \text{I}_2.
\end{split}
\end{equation}
Using the commutator estimate in \cite{Kato}
\[
\sum_{|l|\leq 2} \left\|D^l (fg)-fD^l g\right\|_{L^2} \leq   C\left(\|\nabla f\|_{L^\infty} \left\|D g\right\|_{L^2} +\left\|D^2 f\right\|_{L^2}   \|g\|_{L^\infty} \right),
\]
we have
\begin{equation}\label{bound of I1}
\begin{split}
 \text{I}_1 \leq \sum_{l=0} ^2 \left\|\partial_x^l (u \theta_x)- u  \partial_x^l   \theta_x\right\|_{L^2} \|\theta\|_{H^2} &\leq C\left(\|u_x\|_{L^\infty} \|\theta\|_{H^2} +   \|u\|_{H^2}\|\theta_x\|_{L^\infty} \right)\|\theta\|_{H^2} \\
   &\leq C_{\kappa}   \left(\|u_x\|_{L^\infty}+\|\theta_x\|^{2}_{L^\infty}\right)\|\theta\|_{H^2}^2 +  \kappa\|u\|^{2}_{H^2}.
\end{split}
\end{equation}
And by integration by parts,
\eqn \label{bound of I2}
\text{I}_2 = -\frac{1}{2}\sum_{l=0} ^2\int u \partial_x \left|\partial_x^l \theta\right|^2 dx= \frac{1}{2}\sum_{l=0} ^2\int u_x \left|\partial_x^l \theta \right|^2 dx \leq  C \|u_x\|_{L^\infty} \|\theta\|_{H^2}^2.
\een
Since $\beta\leq \frac{\gamma}{4}$, for a sufficiently small $\kappa>0$
\[
\kappa\|u\|^{2}_{H^2}\leq \frac{1}{2}\left\|\Lambda^{\frac{\gamma}{2}}\theta\right\|_{H^2}^2.
\]
By (\ref{bound of I1}) and (\ref{bound of I2}), we obtain
\eqn \label{Hm bound of w}
\frac{d}{dt} \|\theta\|_{H^2}^2 +\left\|\Lambda^{\frac{\gamma}{2}}\theta\right\|_{H^2}^2 \leq C \left(\|u_x\|_{L^\infty}+\|\theta_x\|^{2}_{L^\infty}\right)\|\theta\|_{H^2}^2 \leq C\|\theta\|_{H^2}^3+ C\|\theta\|_{H^2}^4, \quad \beta\leq \frac{\gamma}{4}
\een
from which we deduce that there is $T=T(\|\theta_{0}\|_{H^{2}})$ such that
\[
\|\theta(t)\|_{H^2}\leq  2 \|\theta_0\|_{H^2} \quad \text{for all} \ t<T.
\]
(\ref{Hm bound of w}) also implies (\ref{blow-up criterion 1}).

To show the uniqueness, let $\theta_{1}$ and $\theta_{2}$ be two solutions of (\ref{singular model d}), and let $\theta=\theta_{1}-\theta_{2}$ and $u=u_{1}-u_{2}$. Then, $(\theta,u)$ satisfies the following equations
\[
\theta_t+u_{1}\theta_x-u\theta_{2x}=-\Lambda^{\gamma}\theta, \quad u=-\mathcal{H}(\partial_{xx} )^{\beta} \theta, \quad \theta(0,x)=0.
\]
By taking the $L^{2}$ product of the equation with $\theta$,
\[
\frac{d}{dt}\|\theta\|^{2}_{L^{2}}+2\left\|\Lambda^{\frac{\gamma}{2}}\theta\right\|^{2}_{L^{2}} \leq C \left(\left\|u_{1x}\right\|_{L^{\infty}} +\left\|\theta_{2x}\right\|_{L^{\infty}}\right)\|\theta\|^{2}_{L^{2}}\leq C \left(\left\|\Lambda^{\frac{\gamma}{2}}\theta_{1}\right\|_{H^{2}} +\left\|\theta_{2}\right\|_{H^{2}}\right)\|\theta\|^{2}_{L^{2}}.
\]
So, $\theta=0$ in $L^{2}$ and thus a solution  is unique. This completes the proof of Theorem \ref{LW and blow-up criterion theorem}.
\end{proof}

Theorem \ref{LW and blow-up criterion theorem} provides a local existence result for $\beta \nearrow \frac{1}{2}$ as $\gamma\nearrow 2$. But, we can increase the range of $\beta$ when we deal with (\ref{singular model d}) directly with $\gamma=2$ because we can do the integration by parts.

\begin{theorem}  \label{LW and blow-up criterion theorem 2}
Let $\gamma=2$ and  $0<\beta<1$. For $\theta_0 \in H^2 (\Bbb R)$ there exists $T=T(\|\theta_0\|_{H^2})$ such that a unique solution of (\ref{singular model d}) exists in $C\left([0, T); H^2 (\Bbb R)\right)$.
\end{theorem}

\begin{proof}
We begin the $L^{2}$ bound:
\[
\frac{1}{2}\frac{d}{dt}\left\|\theta\right\|^{2}_{L^{2}}+ \left\|\theta_{x}\right\|^{2}_{L^{2}}\leq \|\theta\|_{L^{\infty}} \left\|\mathcal{H}(\partial_{xx})^{\beta}\theta\right\|_{L^{2}} \left\|\theta_{x}\right\|_{L^{2}}\leq C\|\theta\|^{3}_{H^{2}}.
\]
We next estimate $\theta_{xx}$.  Indeed, after several integration parts, we have
\[
\frac{1}{2}\frac{d}{dt}\left\|\theta\right\|^{2}_{\dot H^{2}} + \Vert \theta \Vert^{2}_{\dot H^{3}}= -\int \left\{\mathcal{H}(\partial_{xx})^{\beta}\theta_{x}\right\}\theta_{x}\theta_{xxx}dx+\frac{1}{2}\int \left\{\mathcal{H}(\partial_{xx})^{\beta}\theta_{x}\right\}\theta_{xx}\theta_{xx}dx =\text{I}_1+\text{I}_2.
\]
When $0<\beta<1$,
\begin{equation*}
\begin{split}
\left|\text{I}_{1}\right|& \leq \left\|\theta_{x}\right\|_{L^{\infty}}\left\|\mathcal{H}(\partial_{xx} )^{\beta} \theta_{x}\right\|_{L^{2}}\left\|\theta_{xxx}\right\|_{L^{2}} =\left\|\theta_{x}\right\|_{L^{\infty}}\left\|\Lambda^{2\beta+1}\theta \right\|_{L^{2}}\left\|\theta_{xxx}\right\|_{L^{2}}\\
& \leq C\left\|\theta\right\|_{H^{2}} \left\|\theta_{x}\right\|^{1-\beta}_{L^{2}}\left\|\theta_{xxx}\right\|^{1+\beta}_{L^{2}} \leq C\left\|\theta\right\|^{4}_{H^{2}} + C\left\|\theta\right\|^{\frac{4-2\beta}{1-\beta}}_{H^{2}}+\frac{1}{4}\left\|\theta_{xxx}\right\|^{2}_{L^{2}}.
\end{split}
\end{equation*}
And
\[
\left|\text{I}_{2}\right| \leq \left\|\mathcal{H}(\partial_{xx})^{\beta}\theta_{x}\right\|_{L^{2}} \left\|\theta_{xx}\right\|^{2}_{L^{4}} \leq C\left\|\mathcal{H}(\partial_{xx})^{\beta}\theta_{x}\right\|_{L^{2}} \left\|\theta_{xx}\right\|^{\frac{3}{2}}_{L^{2}}\left\|\theta_{xxx}\right\|^{\frac{1}{2}}_{L^{2}} \leq C\left\|\theta\right\|^{4}_{H^{2}} +\frac{1}{4}\left\|\theta_{xxx}\right\|^{2}_{L^{2}}.
\]

Therefore, we obtain
\eqn
\frac{d}{dt}\left\|\theta\right\|^{2}_{H^{2}}+ \left\|\theta_{x}\right\|^{2}_{H^{2}}\leq C\left\|\theta\right\|^{4}_{H^{2}} + C\left\|\theta\right\|^{\frac{4-2\beta}{1-\beta}}_{H^{2}}.
\een
This implies that there exists $T=T(\|\theta_0\|_{H^2})$ such that there exists a unique solution of (\ref{singular model d}) in $C\left([0, T); H^2 (\Bbb R)\right)$.
\end{proof}

We may lower the regularity of the initial data to prove a local existence result of a weak solution by considering initial data in $\dot H^{\frac{1}{2}}$. The main tools to achieve this will be the use of the Hardy-BMO duality together with interpolation arguments. However, in order to simplify the computation, we consider an equivalent equation by changing the sign of the nonlinearity:
\begin{subequations} \label{singular model d}
\begin{align}
& \theta_{t} + \left(\mathcal{H}(-\partial_{xx})^{\beta}\theta\right)\theta_{x} +\Lambda^{\gamma}\theta=0, \\
& \theta(0,x)=\theta_{0}(x)
\end{align}
\end{subequations}
This can be obtained from (\ref{singular model d}) via $\theta \mapsto -\theta$. For this equation, we do $\dot H^{\frac{1}{2}}$ estimates and prove that there exists a local existence of a unique solution in that special case.

\begin{theorem} \label{GW}
Let $\gamma=2$ and $0<\beta<\frac{1}{2}$. For any  $\theta_0 \in \dot H^{\frac{1}{2}} (\Bbb R)$, there exists $T=T(\Vert \theta_{0} \Vert_{\dot H^{\frac{1}{2}}})$ such that there exists a unique local-in-time solution in $C([0, T); \dot H^{\frac{1}{2}} (\Bbb R)) \cap L^2\left([0, T); H^{\frac{3}{2}} (\Bbb R)\right)$.
\end{theorem}

\begin{proof}
By recalling that $\Lambda^{2\beta}=(-\partial_{xx})^{\beta}$ we get
\begin{equation*}
\begin{split}
\frac{1}{2} \frac{d}{dt} \Vert \theta \Vert^{2}_{\dot H^{\frac{1}{2}}} +\left\Vert \Lambda^{\frac{1+\gamma}{2}} \theta \right\Vert^{2}_{L^{2}} &=- \int \Lambda^{\frac{1}{2}} \theta \Lambda^{\frac{1}{2}}\left\{\left(\mathcal{H}(-\partial_{xx})^{\beta}\theta\right)\theta_{x} \right\} dx  \\
&=- \int \theta_{x}\Lambda \theta \ \mathcal{H}(-\partial_{xx})^{\beta}\theta  dx=- \int \theta_{x} \mathcal{H} \theta_{x}  \mathcal{H}(-\partial_{xx})^{\beta}\theta dx.
\end{split}
\end{equation*}
We now use the $\mathcal{H}^1$-BMO duality to estimate the right hand side of the last equality. By using the estimate \eqref{hardy} and $\dot H^{\frac{1}{2}} \hookrightarrow BMO$, we obtain
\[
\Vert \theta_{x} \mathcal{H} \theta_{x} \Vert_{\mathcal{H}^1} \leq \Vert \theta \Vert^{2}_{\dot H^{1}}, \quad \left\|\mathcal{H}(-\partial_{xx})^{\beta}\theta\right\|_{L^{2}}\leq C\Vert \theta \Vert_{\dot H^{2\beta +\frac{1}{2}}}
\]
and thus we have
\[
\frac{1}{2} \frac{d}{dt} \Vert \theta \Vert^{2}_{\dot H^{\frac{1}{2}}} +\left\Vert \Lambda^{\frac{1+\gamma}{2}} \theta \right\Vert^{2}_{L^{2}} \leq  C\Vert  \theta \Vert^{2}_{\dot H^{1}}    \Vert \theta \Vert_{\dot H^{2\beta +\frac{1}{2}}}.
\]
By fixing $\gamma=2$ and by using the interpolation inequalities
\[
\Vert \theta \Vert^{2}_{\dot H^1} \leq \Vert \theta \Vert_{\dot H^{\frac{3}{2}}}\Vert \theta \Vert_{\dot H^{\frac{1}{2}}}, \quad \Vert \theta \Vert_{\dot H^{2\beta +\frac{1}{2}}} \leq \Vert \theta \Vert^{2\beta}_{\dot H^{\frac{3}{2}}} \Vert \theta \Vert^{1-2\beta}_{\dot H^{\frac{1}{2}}},
\]
where we use $\frac{1}{2}\leq 2\beta+\frac{1}{2} \leq \frac{3}{2}$ for $\beta \in \left(0,\frac{1}{2}\right)$ to get the second inequality. Hence, we obtain
\begin{equation*}
\begin{split}
\frac{1}{2} \frac{d}{dt} \Vert \theta \Vert^{2}_{\dot H^{\frac{1}{2}}} +\left\Vert \Lambda^{\frac{3}{2}} \theta \right\Vert^{2}_{L^{2}} &\leq  \Vert  \theta \Vert^{2}_{\dot H^{1}}    \Vert \theta \Vert_{\dot H^{2\beta + \frac{1}{2}}} \\
& \leq  \Vert \theta \Vert^{1+2\beta}_{\dot H^{\frac{3}{2}}} \Vert \theta \Vert^{2-2\beta}_{\dot H^{\frac{1}{2}}} \leq \frac{1}{2}\Vert \theta \Vert^{2}_{\dot H^{\frac{3}{2}}} + 2\Vert \theta \Vert^{4\frac{1-\beta}{1-2\beta}}_{\dot H^{\frac{1}{2}}},
\end{split}
\end{equation*}
where we use the condition $\beta \in \left(0,\frac{1}{2}\right)$ again to derive the inequality. This implies local existence of a unique solution up to some time $T=T(\Vert \theta_{0} \Vert_{\dot H^{\frac{1}{2}}})$.
\end{proof}

\subsection{Global well-posedness}

We finally deal with (\ref{singular model d}) with $\gamma=2$.

\begin{theorem} \label{GW}
Let $\gamma=2$ and $\beta<\frac{1}{4}$. For any  $\theta_0 \in H^2 (\Bbb R)$, there exists a unique global-in-time solution in $C\left([0, \infty); H^2 (\Bbb R)\right)$.
\end{theorem}

\begin{proof}
By Theorem \ref{LW and blow-up criterion theorem}, we only need to control the quantities in (\ref{blow-up criterion 1}). Let  $u=-\mathcal{H}(\partial_{xx})^{\beta}\theta$. We first note that (\ref{singular model d}) satisfies the maximum principle and so
\[
\left\|\theta(t)\right\|_{L^{\infty}}\leq \left\|\theta_{0}\right\|_{L^{\infty}}\leq C \|\theta_{0}\|_{H^{2}}.
\]

We take the $L^2$ inner product of (\ref{singular model d}) with $\theta$. Then,
\eqn
\frac{1}{2}\frac{d}{dt}\|\theta\|^{2}_{L^{2}} +\|\theta_{x}\|^{2}_{L^{2}}=-\int u\theta_{x}\theta dx\leq \|\theta_{0}\|_{L^{\infty}} \|u\|_{L^{2}}\|\theta_{x}\|_{L^{2}}.
\een
Since
\[
 \|u\|_{L^{2}} \leq C\|\theta\|^{1-2\beta}_{L^{2}} \|\theta_{x}\|^{2\beta}_{L^{2}} \quad \text{for $\beta< \frac{1}{2}$},
\]
we have
\eqn \label{theta L2 bound}
\|\theta(t)\|^{2}_{L^{2}} +\int^{t}_{0}\|\theta_{x}(s)\|^{2}_{L^{2}}ds\leq C\left(t, \|\theta_{0}\|_{H^{2}}\right).
\een

\noindent We next take $\partial_x $ to (\ref{singular model d}), take its $L^2$ inner product with $\theta_x$, and integrate by parts to obtain
\begin{equation*}
\begin{split}
\frac12 \frac{d}{dt}\|\theta_x\|_{L^2} ^2 + \left\|\theta_{xx}\right\|_{L^2}^2 =\int u \theta_x \theta_{xx}dx  \leq 2\|u\|^{2}_{L^{\infty}}\|\theta_{x}\|^{2}_{L^{2}}+\frac{1}{2}\left\|\theta_{xx}\right\|_{L^2}^2.
\end{split}
\end{equation*}
Since
\[
\|u\|^{2}_{L^{\infty}} \leq C\|\theta\|^{2}_{L^{2}}+C\|\theta_{x}\|^{2}_{L^{2}} \quad \text{when $\beta<\frac{1}{4}$},
\]
we obtain
\eqn \label{theta H1 bound}
\|\theta_{x}(t)\|^{2}_{L^{2}} +\int^{t}_{0}\|\theta_{xx}(s)\|^{2}_{L^{2}}ds\leq C\left(t, \|\theta_{0}\|_{L^{1}}, \|\theta_{0}\|_{H^{2}}\right) \quad \text{when $\beta<\frac{1}{4}$.}
\een
By (\ref{theta L2 bound}) and (\ref{theta H1 bound}), we finally obtain
\[
\int^{t}_{0}\left(\|\theta_{x}(s)\|_{L^{\infty}}+ \|u_{x}(s)\|_{L^{\infty}}\right)ds \leq C\int^{t}_{0}\left(\|\theta_{x}(s)\|_{L^{2}}+ \|\theta_{xx}(s)\|_{L^{2}}\right)ds \leq C\left(t, \|\theta_{0}\|_{L^{1}}, \|\theta_{0}\|_{H^{2}}\right)
\]
and so we complete the proof of Theorem \ref{GW}.
\end{proof}

\section{Appendix}

This appendix is briefly written based on \cite{Bahouri}. We first provide notation and definitions in the Littlewood-Paley theory.  Let $\mathcal{C}$ be the ring of center 0, of small radius $\frac{3}{4}$ and great radius $\frac{8}{3}$. We take smooth radial functions $(\chi, \phi)$ with values in $[0,1]$  that are supported on the ball $B_{\frac{4}{3}}(0)$ and $\mathcal{C}$, respectively, and satisfy
\begin{equation}
\begin{split}
& \chi(\xi)+ \sum^{\infty}_{j=0}\phi\left(2^{-j}\xi\right)=1 \ \  \forall \ \xi \in \mathbb{R}^{d},\\
&  \sum^{\infty}_{j=-\infty}\phi\left(2^{-j}\xi\right)=1 \ \  \forall \ \xi \in \mathbb{R}^{d}\setminus\{0\},\\
& \left|j-j^{'}\right|\ge 2  \ \Longrightarrow \ \text{supp}\ \phi\left(2^{-j}\cdot\right)\bigcap \text{supp}\ \phi\left(2^{-j^{'}}\cdot\right)=\emptyset,\\
& j\ge 1  \ \Longrightarrow \ \text{supp}\ \chi \bigcap \text{supp}\ \phi\left(2^{-j}\cdot\right)=\emptyset.
\end{split}
\end{equation}
From now on, we use the notation
\[
\phi_{j}(\xi)=\phi\left(2^{-j}\xi\right).
\]
We define dyadic blocks and lower frequency cut-off functions.
\begin{equation}
\begin{split}
& h=\mathcal{F}^{-1}\phi, \quad \widetilde{h}=\mathcal{F}^{-1}\chi,\\
& \Delta_{j}f=\phi_{j}\left(D\right)f=2^{jd} \int_{\mathbb{R}^{d}} h\left(2^{j}y\right)f(x-y)dy,\\
& S_{j}f=\chi\left(2^{-j}D\right)f=2^{jd} \int_{\mathbb{R}^{d}} \widetilde{h}\left(2^{j}y\right)f(x-y)dy,\\
& \Delta_{-1}f=\chi\left(D\right)f=\int_{\mathbb{R}^{d}} \widetilde{h}\left(y\right)f(x-y)dy.
\end{split}
\end{equation}
Then, the homogeneous Littlewood-Paley decomposition is given by
\[
f=\sum_{j\in \mathbb{Z}} \Delta_{j}f \ \  \text{in} \ \  \mathcal{S}^{'}_{h},
\]
where $\mathcal{S}^{'}_{h}$ is the space of tempered distributions $u\in \mathcal{S}^{'}$ such that
\[
\lim_{j\rightarrow -\infty}S_{j}u=0\quad \text{in $\mathcal{S}'$}.
\]

We now define the homogeneous Besov spaces:
\[
\dot{B}^{s}_{p,q} =\left\{f\in \mathcal{S}^{'}_{h}: \ \left\|f\right\|_{\dot{B}^{s}_{p,q}}=\left\|\left\|2^{js}\left\|\Delta_{j}f\right\|_{L^{p}}\right\|_{l^{q}(\mathbb{Z})} \right\|<\infty \right\}.
\]

We also recall Bernstein's inequality in 1D : for $1\leq p\leq q \leq \infty$ and $k\in \mathbb{N}$,
\eqn\label{bernstein}
\sup_{|\alpha|=k} \left\|\partial^{\alpha}\Delta_{j}f \right\|_{L^{p}} \leq C 2^{jk} \left\|\Delta_{j}f \right\|_{L^{p}}, \quad \left\|\Delta_{j}f \right\|_{L^{q}} \leq C 2^{j\left(\frac{1}{p}-\frac{1}{q}\right)} \left\|\Delta_{j}f \right\|_{L^{p}}.
\een

\section*{Acknowledgments}

H.B. was supported by the National Research Foundation of Korea (NRF-2015R1D1A1A01058892).

R.G.B. was supported by the LABEX MILYON (ANR-10-LABX-0070) of Universit\'e de Lyon, within the program ``Investissements d'Avenir'' (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR), and by the Universidad de Cantabria.

O.L. was partially supported by the Marie-Curie Grant, acronym: TRANSIC, from the FP7-IEF program and  by the ERC through the Starting Grant project H2020-EU.1.1.-63922.

Both O. L. and R.G.B. were partially supported by the Grant MTM2014-59488-P from the former Ministerio de Econom\'ia y Competitividad (MINECO, Spain).