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\title[H\" olderian FCLT for mult-indexed summation processes]{H\" olderian functional limit theorem for multi-indexed summation processes}
\author[Račkauskas, Suquet, Zemlys]{A. Račkauskas\inst{1,3}, Ch. Suquet\inst{2}, V. Zemlys\inst{1,2}}
\institute[Vilnius University] {
    \inst{1}%
    Vilniaus University
    \and
    \inst{2}%
    Universite des Sciences et Technologies de Lille
    \and	
    \inst{3}%
    Institute of Mathematics and Informatics
}


\date{27$^{\text{th}}$ of June, 2006.}

\begin{document}
\begin{frame}
    \titlepage
\end{frame}
\section{Motivation}
\subsection{One dimensional case}
\begin{frame}
   \frametitle{Polygonal line}
   \begin{itemize}
       \item   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*}
\item 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*}
 
   \end{itemize}
\end{frame}

\begin{frame}
    \frametitle{H\" older spaces} 
    \begin{itemize}
	\item $H_\alpha([0,1]^d,\mathbb{R})$ -- space 
	    of continuous functions $x:[0,1]^d\to \mathbb{R}$, satisfying $w_{\alpha}(x,1)<\infty$,
	    where
	    \begin{align*} %\label{f:walpha}
		w_\alpha(x,\delta)=\sup_{\bm{t},\bm{s}\in [0,1]^d,|\bm{t}-\bm{s}|<\delta}
		\dfrac{|x(\bm{t})-x(\bm{s})|}{(|\bm{t}-\bm{s}|)^\alpha}.
	    \end{align*}
	\item $H^o_\alpha([0,1]^d,\mathbb{R})$ subset of
	    $H_\alpha([0,1]^d,\mathbb{R})$ of functions satisfying
	    \begin{align*}
		\lim_{\delta\to0}w_\alpha(x,\delta)=0\
	    \end{align*}
	\item Norm 
	    \begin{align*}
	    	\|x\|_\alpha:=\|x\|_\infty+w_\alpha(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_\alpha([0,1]).
    \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}
\subsection{General summation process}
\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}}(\bm{t})=\sum_{\bm{j}\le \bm{n}}|R_{\bm{n},\bm{j}}|^{-1}|R_{\bm{n},\bm{j}}\cap
		[\bm{0},\bm{t}]|X_{\bm{j}},
	    \end{align*}
	    where
	    \begin{align*}
	    R_{\bm{n},\bm{j}} := [(j_1-1)/n_1, j_1/n_1]\times\cdots\times
[(j_d-1)/n_d, j_d/n_d].
	    \end{align*}
   \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Alternative definitions} 
    \begin{itemize}
	\item Alexander and Pyke (1986)
	    \begin{align*}
		Z_n(A)=\sum_{\bm{j}}|(nA)\cap (\bm{j}-1,\bm{j}]|
	    \end{align*}
%	    where $C_{\bm{j}}=(\bm{j}-\bm{1},\bm{j}]$
	\item Erickson (1981)
	    \begin{align*}
		\zeta_{\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*}
	\item Khoshnevisan (2002)
	    \begin{align*}
		\bm{S}_n(\bm{t})=\int_{[0,n\bm{t}]}\Xi(ds)
	    \end{align*}
	    where 
	    \begin{align*}
		\Xi(s)=|[[\bm{s}],\bm{s}]|X_{[\bm{s}]+\bm{1}}
	    \end{align*}
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Functional central limit theorem}
    \begin{itemize}
	\item Weak convergence of $\{\pi(\bm{n})^{-1/2}\xi_{\bm{n}}(\bm{t}), \bm{n}\in
	    \mathbb{N}^d \} $, 
	    when $m(\bm{n})\to\infty$,  where $m(\bm{n}):=n_1\wedge\ldots\wedge n_d$, $\pi(\bm{n})=\prod n_i$. 
	 \vspace{20pt}

	\item  Weak convergence of $\{n^{-d/2}Z_n(A), n\in \mathbb{N} \}$, when
	    $n\to \infty$.  
	    %\begin{align*}
	   %n^{-d/2}Z_n \xrightarrow{\mathcal{D}}W,	\text{ when } n\to\infty.
	    %\end{align*}
 	 \vspace{20pt}
	\item Weak convergence of $\{\pi(\bm{n})^{-1/2}\zeta_{\bm{n}}(A),
	    \bm{n}\in\mathbb{N}^d \} $, when $m(\bm{n})\to\infty$
 	
     \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Limiting process -- Wiener sheet} 
    \begin{itemize}
	\item  For $\bm{t}\in [0,1]^d$: 
	    \begin{definition}
	    $W(\bm{t})$ -- Gaussian with
	    $EW(\bm{t})=0$ and $EW(\bm{t})W(\bm{s})=(t_1\wedge
	    s_1)\dots(t_d\wedge s_d)$.
	\end{definition}

	\item<2>   For $A$ - Borel subset of $[0,1]^d$: 
	    \begin{definition}
	    $W(A)$ -- Gaussian with $EW(A)=0$ and $EW(A)W(B)=|A\cap B|$, for Borels $A,B\subset [0,1]^d$.
	\end{definition}
	    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Previous results}
     \begin{itemize}
	\item Bass (1985), Alexander, Pyke (1986), continuous function space.
	    Sufficient conditions: $EX_1^2<\infty$ and metric entropy condition
	    on collection of sets.

	\item Erickson (1981), H\" older function space.  Relationship of moment,
	    metric entropy and H\" older  smoothness conditions. For collection
	    $\mathcal{Q}_d=\{[\bm{0},\bm{t}], \bm{t}\in [0,1]^d\}$, the FCLT
	    holds in $H_\alpha([0,1]^d)$, if $E|X_{\bm{1}}|^q<\infty$, for
	    $q>d/(1/2-\alpha)$.

     \end{itemize}
\end{frame}

\section{Our Results}
\begin{frame}
    \frametitle{Main result}
    \begin{align*}
	\pi(\bm{n})^{-1/2}\xi_{\bm{n}} \xrightarrow[m(\bm{n})\to\infty]{\mathcal{D}} W \text{
	in the space } \ab{H}^o\bigl([0,1]^d\bigr),
    \end{align*}
    
    \vspace*{15pt}
    
    if and only if
    
    \vspace*{5pt}
   
    \begin{align*}
	    n_1\dots n_d\mathbf{P}\bigl(|X_{\bm{1}}|>n_1^{1/p}n_2^{1/2}\dots
	    n_d^{1/2}\bigr)\to 0,
	    \text{ when } m(\bm{n}) \to \infty,
	   \end{align*}
	   where $p=1/(1/2-\alpha)$.
\end{frame}
\begin{frame}
    \frametitle{Equivalent condition} 
    Conditions
    \begin{align*}
	    n_1\dots n_d\mathbf{P}\bigl(|X_{\bm{1}}|>n_1^{1/p}n_2^{1/2}\dots
	    n_d^{1/2}\bigr)\to 0, \text{ when } m(\bm{n}) \to \infty.
    \end{align*}
    and
    \begin{align*}
	\sup_{t>0} t^p\mathbf{P}(|X_{\bm{1}}|>t)<\infty
    \end{align*}
    when $d>1$ are equivalent.
\end{frame}
\begin{frame}
    \frametitle{Remarks}
    Erickson's condition:  $E|X_1|^q<\infty$,   when $q>d/(1/2-\alpha)$.
    
    New condition: $ n_1\dots
    n_d\mathbf{P}\bigl(|X_{\bm{1}}|>n_1^{1/p}n_2^{1/2}\dots n_2^{1/2}\bigr)
    \to 0, \text{ when } m(\bm{n}) \to \infty.$ 
    \vspace*{20pt}

    \begin{itemize}
	\item The result of Erickson is improved for collection
	    $\mathcal{Q}_d=\{[\bm{0},\bm{t}], \bm{t}\in [0,1]^2\}$. You
	    ``need'' d moments less.
	    \begin{align*}
		\sup_{t>0}t^p\mathbf{P}(|X_{\bm{1}}|\ge t)<\infty.
	    \end{align*}
	\item If path is fixed: $\bm{n}=(n,\dots,n)$ you need less moments than in one
	    dimensional case
	     \begin{align*}
		\lim_{t\to\infty}t^{\frac{d}{d/2-\alpha}}\mathbf{P}(|X_{\bm{1}}|\ge t)=0.
	    \end{align*}
    \end{itemize}

\end{frame}
\end{document}

