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\usepackage[english]{babel}
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\usepackage{amsmath}
\usepackage{amssymb}
%\usepackage{theorem}
\usepackage{bm}
\usepackage{lmodern}
\usepackage{graphicx}

\newcommand{\ab}[1]{#1_{\alpha}}
\newcommand{\wab}[2][\delta]{w_{\alpha}(#2,#1)}
\newcommand{\normab}[1]{\lVert#1\rVert_{\alpha}}
\newcommand{\eps}{\varepsilon}
\newcommand{\sprod}[1]{\langle #1 \rangle}

%\renewcommand{\theenumi}{\roman{enumi}}
%\renewcommand{\labelenumi}{\theenumi)}
\theoremstyle{change}\newtheorem{teorema}{Proposition}
\theoremstyle{change}\newtheorem{salyga}{}
%	\vspace*{20pt}
%	\vspace*{20pt}

\title{Functional limit theorem for double-indexed summation processes}
\author{Vaidotas Zemlys\\
A. Ra\v ckauskas
}
\institute[Vilnius University] {
    Vilnius University

    Universit\' e des Sciences et Technologies de Lille
    %Department of Mathematics and Informatics,
    %Department of Econometrical Analysis

    Vaidotas.Zemlys@maf.vu.lt

        }
\date{24$^{\text{th}}$ of March, 2006.}

\begin{document}
\begin{frame}
    \titlepage
\end{frame}
%\section{Summation processes}
%\begin{frame}
%    \frametitle{Darbo esmė}
%    \begin{itemize}
%	\item   Papildytas Erickson rezultatas  
%	\item   Apibendrintas Račkausko ir Suquet rezultatas
%
%	\end{itemize}
%\end{frame}
%\section{Motivation}
%\subsection{Summation processes}
\begin{frame}
   \frametitle{Polygonal line}
    Usual polygonal line process
    \begin{align*}
	\xi_n(t)=S_{[nt]}+(nt-[nt])X_{[nt]+1},\quad t\in[0,1],
    \end{align*}
    where
    \begin{align*}
	S_k=X_1+X_2+\ldots+X_k, \quad k=1,2,\ldots
    \end{align*}
\end{frame}

\begin{frame}
    \frametitle{Polygonal line} 
    \includegraphics[scale=1]{W10.ps}
\end{frame}
\begin{frame}
    \frametitle{Donsker-Prokhorov theorem} 
   	\begin{align*}
		n^{-1/2}\xi_n\xrightarrow{\mathcal{D}}W \text{ in the space} C[0,1] \Leftrightarrow EX_1^2<\infty,
    \end{align*}
	 \vspace*{10pt}
	 where $W(t)$ - Wiener process or Brownian motion.
	\vspace*{10pt}
	\begin{itemize}
		\item $W_0=0$.
     	\vspace*{10pt}
     	\item $W_t\ \sim\ N(0,t)$.
     	\vspace*{10pt}
     	\item $EW_tW_s=t\wedge s$.
     	\vspace*{10pt}
     	%\item Jei $t_0<t_1<t_2,\cdots<t_n$, tai
		%\begin{align*}
	%		W_{t_1}-W_{t_0},W_{t_2}-W_{t_1},\ldots,W_{t_n}-W_{t_{n-1}}
    %		\end{align*}
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{H\" older spaces} 
    \begin{itemize}
	\item $H_\rho([0,1]^d,\mathbb{R})$ -- space 
	    of continuous functions $x:[0,1]^d\to \mathbb{R}$, satisfying $w_{\rho}(x,1)<\infty$,
	    where
	    \begin{align*} %\label{f:walpha}
		w_\rho(x,\delta)=\sup_{\bm{t},\bm{s}\in [0,1]^d,|\bm{t}-\bm{s}|<\delta}
		\dfrac{|x(\bm{t})-x(\bm{s})|}{\rho(|\bm{t}-\bm{s}|)}.
	    \end{align*}
	\item $H^o_\rho([0,1]^d,\mathbb{R})$ subset of
	    $H_\rho([0,1]^d,\mathbb{R})$ of functions satisfying
	    \begin{align*}
		\lim_{\delta\to0}w_\rho(x,\delta)=0\
	    \end{align*}
	\item Norm 
	    \begin{align*}
	    	\|x\|_\rho:=\|x\|_\infty+w_\rho(x,1)
	    \end{align*}
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Generalisation of D-P} 
    \begin{align*}
		n^{-1/2}\xi_n\xrightarrow[n\to\infty]{\mathcal{D}}W, \text{ in
		the space } {H}^o_\rho([0,1]),\; \rho(h)=h^\alpha
    \end{align*}
	\begin{itemize}
		\item Lamperti (1962) proved sufficient condition: 
		\begin{align*}
			E|X_1|^p<\infty,
		\end{align*}
		wherre $p>1/(1/2-\alpha)$ and  $0<\alpha<1/2$.
		\pause
		\item Ra\v ckauskas and Suquet (2001) proved
		    necessary and sufficient condition:
	    \begin{align*}
		\lim_{t\to\infty}t^p\mathbf{P}(|X_1|\ge t)=0 \text{ with } p=\frac{1}{1/2-\alpha}.
	    \end{align*}
	\end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Multi-dimensional case}
    %\vspace*{20pt}
    \begin{itemize}
	\item ($X_{\bm{j}}$, $\bm{j}\in \mathbb{N}^d$) -- zero mean independent
	    real random variables. 
	    \vspace*{10pt}
	\item $\bm{n}=(n_1,\ldots,n_d)$, $A$ - Borel subset of $[0,1]^d$ 
	    \begin{align*}
	    \xi_{\bm{n}}(A)=\sum_{\bm{j}\le \bm{n}}|R_{\bm{n},\bm{j}}|^{-1}|R_{\bm{n},\bm{j}}\cap
	  A|X_{\bm{j}},
	    \end{align*}
	    where
	    \begin{align*}
	    R_{\bm{n},\bm{j}} := [(j_1-1)/n_1, j_1/ni_1]\times\cdots\times
[(j_d-1)/n_d, j_d/n_d].
	    \end{align*}
   \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Two dimensional case $A=[0,t_1]\times[0,t_2]$}
    \includegraphics[scale=1]{net.ps}
\end{frame}
\begin{frame}
    \frametitle{Two dimensional case $A=[0,t_1]\times[0,t_2]$}
    \includegraphics[scale=1]{W10x10.ps}
\end{frame}

%\subsection{Functional central limit theorem}
\begin{frame}
    \frametitle{Functional central limit theorem}
    \begin{itemize}
	\item 
	 Conditions for
	    \begin{align*}
	    (n_1\cdots n_d)^{-1/2}\xi_{\bm{n}}\xrightarrow{\mathcal{D}}W,
		\text{ when } m(\bm{n})\to\infty.
	 \end{align*}
         to take place in apropriate space. $m(\bm{n}):=n_1\wedge\ldots\wedge n_d$. 
	 \vspace{20pt}
    \item   $W$ -- Wiener sheet.
	$W(A)$ -- Gaussian with $EW(A)=0$ and $EW(A)W(B)=|A\cap B|$, for Borels
	$A,B\subset [0,1]^d$.


	    
    \end{itemize}
      
\end{frame}

\begin{frame}
    \frametitle{Continuous function space}
     \begin{itemize}
	\item Bass (1985), Alexander, Pyke (1986),
	    sufficient conditions $EX_1^2<\infty$ and
	    \begin{align*}
		\int_0^1\sqrt{\log(N_I(\eps,\mathcal{A},\delta))}d\eps <\infty
	    \end{align*}
	    where $N_I(\eps,\mathcal{A},\delta)$ - metric entropy with
	    inclusion.
     \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{H\" older space}
    %\vspace*{40pt}
    \begin{itemize}
	\item Erickson (1981). Let $E|X_1|^p<\infty$, $r_1(\mathcal{A})\le r$,
	    and $q:=p/2-r>0$. 
	       
	    Then FCLT in $H_\rho(\mathcal{A},\mathbb{R})$, with norm $\|x\|_f$,
	    $\rho=o(f)$ holds for $\rho(h)^p=h^{q}|\log h||\log\log h|^{\beta}$,
	    for each $\beta>1$.
	
    %\item 
    \pause
	\begin{salyga}
	    For $\mathcal{A}:=\{[0,t_1]\times\dots\times[0,t_d]\}$, FCLT holds in
	space $\ab{H}^o([0,1]^d)$ if  $E|X|^q<\infty$, for $q>\frac{d}{1/2-\alpha}$, $0<\alpha<1/2$.
    \end{salyga}
     \end{itemize}
\end{frame}

%\subsection{H\"older space}
%\begin{frame}
%    \frametitle{Multidimensional case} 
%     Erickson (1981) proved that in space $\ab{H}^o\bigl([0,1]^d\bigr)$
%	\begin{align*}
%	    	(n_1\cdots n_d)^{-1/2}\xi_n\xrightarrow{\mathcal{D}}W_d,
%		\text{ when } \bm{n}\to\infty,
%	\end{align*}
%	if
%	\begin{align*}
%		E|X_1|^q<\infty, \text{ when }  q>d/(1/2-\alpha).
%	\end{align*}

%\end{frame}
%\section{Main result}
%\subsection{Formulation}
\begin{frame}
    \frametitle{Main result}
    \begin{align*}
	(n_1n_2)^{-1/2}\xi_{\bm{n}} \xrightarrow[n_1\wedge n_2 \to
	\infty]{\mathcal{D}} W \text{ in the space } \ab{H}^o\bigl([0,1]^2\bigr),
    \end{align*}
    
    \vspace*{15pt}
    
    if and only if
    
    \vspace*{5pt}
   
    \begin{align*}
	    n_1n_2\mathbf{P}\bigl(|X_{\bm{1}}|>n_1^{1/p}n_2^{1/2}\bigr)\to 0,
	    \text{ when } n_1\wedge n_2 \to \infty,
	   \end{align*}
	   where $p=1/(1/2-\alpha)$.
\end{frame}

\begin{frame}
    \frametitle{Remarks}
    Erickson's condition:  $E|X_1|^q<\infty$,   when $q>2/(1/2-\alpha)$.
    
    New condition: $ t_1t_2\mathbf{P}\bigl(|X_{\bm{1}}|>t_1^{1/p}t_2^{1/2}\bigr)
    \to 0, \text{ when } t_1\wedge t_2 \to \infty.$ 
    \vspace*{20pt}

    \begin{itemize}
	\item The result of Erickson is improved for two-dimensional case. You
	    ``need'' twice as less moments.
	    \begin{align*}
		\lim_{t\to\infty}t^p\mathbf{P}(|X_{\bm{1}}|\ge t)=0.
	    \end{align*}
	\item If path is fixed: $\bm{n}=(n,n)$ you need less moments than in one
	    dimensional case
	     \begin{align*}
		\lim_{t\to\infty}t^{\frac{2}{1-\alpha}}\mathbf{P}(|X_{\bm{1}}|\ge t)=0.
	    \end{align*}
    \end{itemize}

\end{frame}
\begin{frame}
    \frametitle{The simulation with one moment} 
    \includegraphics{W60x601.ps}
\end{frame}
\begin{frame}
    \frametitle{The simulation with two moments} 
    \includegraphics{W60x602.ps}
\end{frame}
\begin{frame}
    \frametitle{The simulation with three moments} 
    \includegraphics{W60x603.ps}
\end{frame}
\begin{frame}
    \frametitle{The simulation with all moments} 
    \includegraphics{W60x60r.ps}
\end{frame}

%\subsection{Details of the proof}
\begin{frame}
    \frametitle{Details of the proof} 
    \begin{itemize}
	\item $(\xi_{\bm{n}}, \bm{n}\in\mathbb{N}^2)$ is a net with directed set
	    $\mathbb{N}^2$ and relation $\bm{j}\le \bm{n}\Leftrightarrow j_i\le n_i$,	    $i=1,\ldots,d$.
	    \vspace{20pt}
	    \pause
	\item The finite dimensional distributions of process
	    $(n_1n_2)^{-1/2}\xi_{\bm{n}}(\bm{t})$ converge to those of Wiener
	    sheet, when $n_1\wedge n_2\to \infty$. (Erickson (1981))
\end{itemize}
\end{frame}

\begin{frame}
    \frametitle{Convergence criteria} 
      Let $\{\zeta_{\bm{n}}, \bm{n}\in\mathbb{N}^2\}$ be  net of
    $\ab{H}^o\bigl([0,1]^2\bigr)$ random elements satisfying:

\begin{enumerate}
    \item Finite-dimensional distributions of $\zeta_{\bm{n}}$ converges weakly
	to some $\ab{H}^o\bigl([0,1]^d\bigr)$ random element $\zeta$
	finite-dimensional distributions, when  $n_1\wedge n_2\to\infty$.
    \item $\lim_{a\to \infty}\sup_{\bm{n}} P(\sup_{\bm{t}\in
        T}|\zeta_{\bm{n}}(\bm{t})|>a)=0$
    \item  For each $\eps>0$
        \begin{align*}
	    \lim_{J\to \infty} \limsup_{\bm{n} \to\infty}P(\sup_{j\ge J}2^{\alpha j}\max_{\bm{v}\in
        V_j}|\lambda_{j,\bm{v}}(\zeta_{\bm{n}})|>\eps) = 0.
        \end{align*}
\end{enumerate}
\end{frame}
\begin{frame}
    \frametitle{Definitions} 
    	For each $x\in \ab{H}^o([0,1]^d)$:
\begin{align*}
    \lambda_{0,\bm{v}}(x)&=x(\bm{v}),\quad \bm{v}\in V_0,\notag \\
    \lambda_{j,\bm{v}}(x)&=x(\bm{v})-\dfrac{1}{2}(x(\bm{v}^-)+x(\bm{v}^+)),\quad
    \bm{v}\in V_j,\ \ j\ge 1,
\end{align*}
where $V_j=W_j\backslash W_{j-1}$, $W_j=\{k2^{-j};0\le k\le 2^j\}^2$.
\begin{align*}
  v_i^\pm=
  \begin{cases}
    v_i\pm2^j, & \text{for noneven}\; k_i;  \\
     v_i,& \text{for even}\; k_i.
  \end{cases}
\end{align*}    
\end{frame}
\begin{frame}
    \frametitle{Prokhorov theorem} 
    Let $(X_\alpha,\alpha\in I)$ be a net of separable space random elements. 
    If $(X_\alpha)$ is asymptoticaly tight, i.e. for each
    $\eps>0$ there exists a compact $K_\eps$ such that
\begin{align*}
    \liminf_{\alpha}P(X_\alpha \in K_\eps)>1-\eps,
\end{align*}
then net $(X_\alpha)$ is relatively compact, (1.3.9 theorem, p. 21
psl. van der Vaart ir Wellner (1996))
\end{frame}
\begin{frame}
    \frametitle{Arzela-Ascoli theorem}
     The subset $A$ of $\ab{H}^o$ is relatively compact if and only if
    \begin{align*}
    &\sup_{x\in A}\sup_{t\in T}|x(t)|<\infty  \\%label{AApirma:salyga}\\
    \intertext{and}
    &\lim_{J\to\infty}\sup_{x\in A}\sup_{j\ge J}\max_{v\in V_j}2^{\alpha
    j}|\lambda_{j,v}(x)|=0.%\label{AAantra:salyga}
\end{align*}
\end{frame}
\begin{frame}
    \frametitle{Necessity}
	We have
\begin{align*}
    X_{\bm{k}}&=(S_{(k_1,k_2)}-S_{(k_1-1,k_2)})-(S_{(k_1,k_2-1)}-S_{(k_1-1,k_2-1)}),\\
\end{align*}
thus	
\begin{align*}
    &P(n_1^{-1/p}  n_2^{-1/2} \max_{{\bm{1}}\le \bm{k}\le
    \bm{n}}|X_{\bm{k}}|>t)   \\
	& \le
    P\Bigl(2(n_1n_2)^{1/2}\max_{|\frac{\bm{k}-\bm{l}}{\bm{n}}|=
    |\frac{\bm{1}}{\bm{n}}|}\dfrac{|S_{\bm{k}}-S_{\bm{l}}|}{
    |(\bm{k}-\bm{l})/\bm{n}|^\alpha}>t\Bigr)\nonumber\\
 & \le P(\wab{(n_1n_2)^{1/2}\xi_{\bm{n}}}>t/2)
\end{align*}
\end{frame}
\begin{frame}
\frametitle{Sufficiency}
	 Because of the symmetry it suffices to check that for every $\eps>0$ 
	\begin{align*}
    \lim_{J\to\infty}\limsup_{\bm{n}\to\infty}P\biggl(\sup_{j\ge J}2^{\alpha
     j}(n_1n_2)^{-1/2}\max_{\substack{0\le k\le 2^j-1\\ 0\le
     \ell \le
     2^j}}\Delta_{\bm{n}}(t_{k+1}, t_k; s_\ell)
     >\eps\biggr)=0,
\end{align*}
where $t_k=k2^{-j}$ and $s_\ell=\ell 2^{-j}$.
	
\end{frame}

\begin{frame}
    \frametitle{Sufficiency}
	For difference $\Delta_{\bm{n}}(t, t'; s)=|\xi_{(n_1, n_2)}(t',
	s)-\xi_{(n_1,n_2)}(t, s)|$, we have
	
\begin{align*}
    \sup_{s\in [0, 1]}\Delta_{\bm{n}}(t, t'; s)&\le 3\chi\{t'-t\ge 1/n_1\}
    \psi_{\bm{n}}(t',t)\\
	&+6\min\big\{1, n_1(t'-t)\big\}\zeta_{n_1,
n_2},
\end{align*}
where
\begin{align*}
    \psi_{\bm{n}}(t',t)&= \max_{1\le k\le n_2}
    \Big|\sum_{i=[n_1t]+1}^{[n_1t']}S^{(1)}_{(i,\ k)}\Big|,\\
 \zeta_{n_1, n_2}&=\max_{1\le i\le
 n_1}\max_{1\le k\le n_2}|S^{(1)}_{(i,\ k)}|.
 \end{align*}
\end{frame}

\end{document}

\subsection{Wiener sheet}
\begin{frame}
    \frametitle{Wiener sheet}
      \begin{itemize}
	\item $W(\bm{t})=0$,  $t_i=0$, for some $1\le i\le d$.
	    \vspace{10pt}
	\item $W_d(\bm{t})\ \sim\ N(0,t_1\cdots t_d)$,
	    $\bm{t}=(t_1,\ldots,t_d)$.
	    \vspace{10pt}
	\item $EW_d(\bm{t})W_d(\bm{s})=(t_1\wedge
	    s_1)\cdots(t_d\wedge s_d)$.
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Basic properties} 
     \begin{itemize}
	\item By fixing  $t_i$, you get $d-1$ dimensional Wiener sheet.
	    \vspace{10pt}
	\item For non overlaping rectangles, independent random variables. $EW(\bm{s})(W(\bm{t})-W(\bm{s}))=0$, $\bm{s}\le\bm{t}.$
	    \vspace{10pt}
	\item Trajectories are in $\ab{H}^o\bigl([0,1]^d\bigr)$, $0<\alpha<1/2$.
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Non overlaping rectangles}
    \includegraphics{Wnet.ps}
\end{frame}
