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\usepackage[english]{babel}
\usepackage[utf8x]{inputenc}
\usepackage[L7x]{fontenc}
\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}
\DeclareMathOperator{\diam}{diam}
%\renewcommand{\theenumi}{\roman{enumi}}
%\renewcommand{\labelenumi}{\theenumi)}
\theoremstyle{change}\newtheorem{teorema}{Teiginys}
\theoremstyle{change}\newtheorem{salyga}{}
%	\vspace*{20pt}
%	\vspace*{20pt}

\title[FCRT daugiamačio indekso procesams]{Funkcinė centrinė ribinė teorema daugiamačio indekso sumavimo procesams}
%\shorttitle{FCRT daugiamačio indekso procesams}
\author[Račkauskas, Suquet, Zemlys]{A. Račkauskas\inst{1,3}, Ch. Suquet\inst{2}, V. Zemlys\inst{1,2}}
\institute[Vilnius University] {
    \inst{1}%
    Vilniaus Universitetas
    \and
    \inst{2}%
    Universite des Sciences et Technologies de Lille
    \and	
    \inst{3}%
    Matematikos ir informatikos institutas
}
\date{2006 gegužės 23}

\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{Apibrėžimai}
%\subsection{Sumavimo procesai}
\begin{frame}
    \frametitle{Bendriausia sumavimo proceso konstrukcija}
    %\vspace*{20pt}
    \begin{itemize}
	\item ($X_{\bm{j}}$, $\bm{j}\in \mathbb{N}^d$) -- nepriklausomi vienodai pasiskirstę 	    atsitiktiniai dydžiai su nuliniu vidurkiu. 
	    \vspace*{10pt}
	\item $\bm{n}=(n_1,\ldots,n_d)$, $A$ -- $[0,1]^d$ Borelio poaibis 
	    \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*}
	    čia
	    \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{Dvimatis atvejis $A=[0,t_1]\times[0,t_2]$}
    \includegraphics[scale=1]{net.ps}
\end{frame}
\begin{frame}
    \frametitle{Dvimatis atvejis $A=[0,t_1]\times[0,t_2]$}
    \includegraphics[scale=1]{W10x10.ps}
\end{frame}

%\begin{frame}
%    \frametitle{Vienmačio atvejo apibendrinimas}
%    %\vspace*{20pt}
%       \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
%	  A|X_{\bm{j}},
%	 \end{align*}
%    \begin{itemize}
%	\item Imant $A=[0,t]$ susiveda į vienmatį atvejį
%    \end{itemize}
%\end{frame}

\begin{frame}
    \frametitle{Funkcinė centrinė ribinė teorema} 
     Funkcinė centrinė ribinė teorema (FCRT) galioja, jei
    \begin{align*}
	    (n_1\cdots n_d)^{-1/2}\xi_{\bm{n}}\xrightarrow{\mathcal{D}}W,
		\text{ kai } m(\bm{n})\to\infty.
	\end{align*}
	tam tikroje trajektorijų erdvėje,    čia $m(\bm{n}):=n_1\wedge\ldots\wedge n_d$. 

\end{frame}

\begin{frame}
    \frametitle{Trajektorijų erdvės} 
    \uncover<1->{
    $(T,d)$ -- parametrų aibė, 	$(B,\|\|)$ -- reikšmių aibė.
    }
    \begin{itemize}
	\item<2,4-> $C(T,B)$ -- tolydžių funkcijų aibė su įprastine norma 
	    \begin{align*}
		\|x\|_\infty=\sup_{t\in T}\|x\|
	    \end{align*}	
	\pause
	
	\item<3,4-> $H_\rho(T,B)$ -- aibė tolydžių funkcijų tenkinančių $w(x,1)<\infty$, čia
	    \begin{align*}
		w_\rho(x,\delta)=\sup_{t,s\in T,d(t,s)<\delta}
		\dfrac{\|x(t)-x(s)\|}{\rho(d(t,s))}.
	    \end{align*}
	    su norma
	    
	    \begin{align*}
	    	\|x\|_\rho:=\|x\|_\infty+w_\rho(x,1)
	    \end{align*}
    \end{itemize}
\end{frame}

\begin{frame}
   \frametitle{Vienmatis atvejis}
    
   
    \only<1>{$\bm{n}=(n_1,\ldots,n_d)$, $A$ -- $[0,1]^d$ Borelio poaibis 
	    \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
	  A|X_{\bm{j}},
	    \end{align*}

	    $A=[0,t]$, $\bm{n}=(n)$.    }
    \only<2>{
    \begin{align*}
	\xi_n(t)=S_{[nt]}+(nt-[nt])X_{[nt]+1},\quad t\in[0,1],
    \end{align*}
    čia
    \begin{align*}
	S_k=X_1+X_2+\ldots+X_k, \quad k=1,2,\ldots
    \end{align*}}
\end{frame}

\begin{frame}
    \frametitle{Laužtė} 
    \includegraphics[scale=1]{W10.ps}
\end{frame}

\begin{frame}
    \frametitle{Rezultatai realiems a. d.}
    \begin{itemize}
	\item Erdvėje $C[0,1]$, Donskerio-Prochorovo invariantiškumo principas.
	    Būtina ir pakankama sąlyga $EX_1^2<\infty$
	\item Erdvėje $H^o_\alpha([0,1])$, Lamperti (1962), pakankama sąlyga
	    $E|X_1|^q<\infty$, $q>1/(1/2-\alpha)$, $0<\alpha<1/2$.
    	
	\item Erdvėje $H^o_\alpha([0,1])$, Račkauskas-Suquet. Būtina ir
	    pakankama sąlyga
	    \begin{align*}
		\lim_{t\to\infty}t^{1/(1/2-\alpha)}P(|X_1|>t)=0
	    \end{align*}
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Rezultatai vektoriniams a.d.}
    \begin{itemize}
	\item Kuelbs (1973). $C([0,1],B)$ -- būtina ir pakankama sąlyga $X_1\in CRT(B)$.
	\item Račkauskas ir Suquet (2003). $H_\rho^o([0,1],B)$ -- būtina ir
	    pakankama sąlyga $X_1\in CRT(B)$ ir kiekvienam $A>0$
	    \begin{align*}
		\lim_{t\to\infty}tP(\|X_1\|>A\theta(t))=0,
	    \end{align*}
	    čia $\theta(t)=t^{1/2}\rho(1/t)$.
    \end{itemize}
\end{frame}

\begin{frame}
    \frametitle{Ribinis procesas} 
    \begin{itemize}
	\item  $W$ -- Wynerio paklodė (sheet).	
	    $W(A)$ -- normalusis su $EW(A)=0$ ir $EW(A)W(B)=|A\cap B|$, Borelio
	    aibėms $A,B\subset [0,1]^d$.
	\item Imant $A=[0,t]\subset[0,1]$ įprastas Brauno judesys
    \end{itemize}
\end{frame}

\begin{frame}
    \frametitle{Entropinė sąlyga} 
    $\{X(t), t\in T\}$ - normalusis procesas. $d(t,s)=\sqrt{E|X(t)-X(s)|^2}$ 	
    \begin{teorema}
	%\item Egzistavimo klausimas susiveda į Gausinio proceso su parametru iš
	%    abstrakčios erdvės $(T,d)$ egzistavimą.
	 Dudley (1973). Tegu $N(\eps)=\exp(H(\eps))$ yra mažiausias
	    $n$ toks, kad egzistuoja aibės $A_1,\dots,A_n\subset T$, tokios, kad
	    $T\subset \cup_1^nA_k$ ir $\diam A_k\le 2\eps$ visiems $k$. Tegu
		\begin{align*}
		    f(t):=\int_0^t(H(s))^{1/2}ds<\infty
		\end{align*}
		Tada $X(t)$ turi modifikacijas $C(T,\mathbb{R})$ ir
		$H_f(T,\mathbb{R})$ erdvėse.
    \end{teorema}    
\end{frame}

\begin{frame}
    \frametitle{Vynerio paklodės trajektorijų savybės} 
    \begin{itemize}
	\item Parametrų aibė $(\mathcal{A}:=\mathcal{B}([0,1]^d),d)$ su 	
	         \begin{align*}
		     d(A,B)=\sqrt{|A\Delta B|}=\sqrt{{E(W(A)-W(B))^2}}
	     	 \end{align*}
	     \item Remiantis Dudley teorema, $W\in C(\mathcal{A},\mathbb{R})$ ir
		 $W\in H_f(\mathcal{A},\mathbb{R})$.
    \end{itemize}
\end{frame}

\begin{frame}
    \frametitle{FCRT tolydžių funkcijų erdvėje} 
     FCRT-$C(\mathcal{A},\mathbb{R})$.  Bass (1985), Pyke (1986),
	    pakankama sąlyga $EX_1^2<\infty$ ir
	    \begin{align*}
		\int_0^1\log(N_I(\eps,\mathcal{A},\delta)d\eps <\infty
	    \end{align*}
	    čia $N_I(\eps,\mathcal{A},\delta)$ - metrinė entropija su įdėjimu.
\end{frame}

\begin{frame}
    \frametitle{Hiolderiškos trajektorijos} 
    \begin{itemize}
	\item<1-2>  Jei $T$ - visiškai aprėžta, $H_\rho(T,\mathbb{R})$ yra
	      $\sigma$-kompaktiška separabili tiesinė erdvė su norma $\|x\|_g$,
	      kai  $\rho=o(g)$.

	\item<2> Erickson (1981). Tegu $E|X_1|^p<\infty$, $r_1(\mathcal{A})\le r$,
	    ir $q:=p/2-r>0$. 
	 \begin{salyga}	    
	    Tada FCRT erdvėje $H_\rho(\mathcal{A},\mathbb{R})$ su norma $\|x\|_f$,
	    $\rho=o(f)$ galioja su $\rho(h)^p=h^{q}|\log h||\log\log
	    h|^{\beta}$, kiekvienam $\beta>1$.
	\end{salyga}
    \item<3> Imant $\mathcal{A}:=\{[0,t_1]\times\dots\times[0,t_d],
	(t_1,\dots,t_d)\in [0,1]^d\}$, pakankama sąlyga supaprastėja:
	\begin{salyga}   
	    FCRT galioja erdvėje $H_\alpha([0,1]^d,\mathbb{R})$, jei 
	    $E|X|^q<\infty$, kai $q>\frac{d}{1/2-\alpha}$, $0<\alpha<1/2$.
        \end{salyga}
     \end{itemize}
\end{frame}

\begin{frame}
    \frametitle{Motyvacija} 
    \begin{itemize}
	\item<1-> Apibendrinti Račkausko-Suquet rezultatus
	    \begin{itemize}
		\item<2-> FCRT-$H^o_\alpha([0,1]^d,\mathbb{R})$
		\item<3-> FCRT-$H^o_\alpha([0,1]^d,\mathbb{H})$
	    \end{itemize}
   
	\item<4-> Rasti būtinas ir pakankamas sąlygas FCRT.
    \end{itemize}
\end{frame}

\begin{frame}
    \frametitle{Ankstesni rezultatai} 
    \begin{itemize}
	\item<1-> FCRT-$H^o_\alpha([0,1]^2,\mathbb{R})$
	    \begin{itemize}
		\item<2-> Magistrinis darbas (2004)
		\item<3-> Pranešimas LMD konferencijoje(2004)	
		\item<4-> Straipsnis LMR (2005)
		\item<5-> Pranešimas SPRF konferencijoje (2006)
	    \end{itemize}
	\item<6-> FCRT-$H^o_\alpha([0,1]^d,\mathbb{R}$, pranešimas LMD
	    konferencijoje (2005).
    \end{itemize}
\end{frame}

\begin{frame}
    \frametitle{Paskelbtas rezultatas}
    \begin{align*}
	(n_1n_2)^{-1/2}\xi_{\bm{n}} \xrightarrow[n_1\wedge n_2 \to
	\infty]{\mathcal{D}} W \text{ erdvėje } \ab{H}^o\bigl([0,1]^2\bigr),
    \end{align*}
    
    \vspace*{15pt}
    
    tada ir tik tada, kai
    
    \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{ kai } n_1\wedge n_2 \to \infty,
	   \end{align*}
	   čia $p=1/(1/2-\alpha)$.
\end{frame}

\begin{frame}
    \frametitle{Separabilumas, kai $T=[0,1]^d$} 
   
    $H^o_\rho([0,1]^d)$ -- erdvės $H_\rho([0,1]^d)$ poerdvis, kurio elementams galioja
    \begin{align*}
		\lim_{\delta\to\infty}w_\rho(x,\delta)=0\
    \end{align*}
        
    \begin{teorema}
	 Račkauskas-Suquet (2000). Su $\rho=h^\alpha\log^\beta(c/h)$, erdvė $H_\rho^o([0,1]^d,B)$ yra
	    separabili Banacho erdvė, kuriai egzistuoja Schauder išdėstymas.
%	      $$H_\rho^o([0,1]^d,B):=\{x\in  H_\rho([0,1]^d,B):\lim_{\delta\to\infty}w_\rho(x,\delta)=0\}$$.
	\end{teorema}
\end{frame}

%\begin{frame}
%    \frametitle{ Dvimatis atvejis $A=[0,t_1]\times[0,t_2]$}
%    \includegraphics[scale=1]{W10x10.ps}
%\end{frame}

\begin{frame}
    \frametitle{Apibrėžimai} 
    Kiekvienai funkcijai $x\in {H}^o_\alpha(\mathbb{H})$:
    \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*}
    čia $V_j=W_j\backslash W_{j-1}$, $W_j=\{k2^{-j};0\le k\le 2^j\}^d$.
\begin{align*}
  v_i^\pm=
  \begin{cases}
    v_i\pm2^j, & \text{kai}\; k_i\; \text{yra nelyginis};  \\
     v_i,& \text{kai}\; k_i\; \text{yra lyginis}.
  \end{cases}
\end{align*}    
\end{frame}
\begin{frame}
    \frametitle{Schauder išdėstymas}
    \begin{align*}
	x=\sum_{j=0}^\infty (E_j-E_{j-1})x,
    \end{align*}
    čia
    \begin{align*}
	E_jx=\sum_{i=0}^j\sum_{\bm{v}\in
	V_i}\lambda_{i,\bm{v}}(x)\Lambda_{i,\bm{v}}
    \end{align*}
    Piramidalinės funkcijos $\Lambda_{j,v}$ apibrėžiamos kaip
%obtained by dyadic translations and changes of scale:
    $$
    \Lambda_{j,\bm{v}}(\bm{t})=\Lambda(2^j(\bm{t}-\bm{v})),\quad \bm{t}\in [0,1]^d,
    $$
čia
$$
\Lambda(\bm{s}):=
\max\bigl(0,\;1-\max_{s_i<0}|s_i|-\max_{s_i>0}s_i\bigr),\quad
\bm{s}=(s_1,\ldots,s_d)\in[-1,1]^d.
$$
    
\end{frame}
%\begin{frame}
%    \frametitle{Schauder išdėstymas} 
%    \begin{teorema}\label{Prop7}
%    Erdvei $\ab{H}^o(\mathbb{H})$ egzistuoja Schauder dekompozicija
%    $$
%    H^o_\alpha(\mathbb{H})=\bigoplus_{i=0}^\infty \bm{W}_i,
%    $$
%    čia $\bm{W}_0$ yra $\mathbb{H}$-funkcijų erdvė, tolydžių $[0,1]^d$ ir
%    affinių   kiekviename $T_0$ simplekse $\Delta$. Kiekvienam $i\geq 1$, $\bm{W}_i$ yra
%    $\mathbb{H}$-funkcijų erdvė tolydžių $[0,1]$ ir affinių kiekviename $T_i$ simplekse $\Delta$ 
%    $T_i$ bei  virstančių nuliu kiekvienoje viršūnėje iš $W_{i-1}$.
%\end{teorema}
%\end{frame}
\begin{frame}
    \frametitle{Tirštumo kriterijus} 
   \begin{teorema}\label{t:tight}
    Tegu $\{\phi_{\bm{n}}, \bm{n}\in\mathbb{N}^d\}$ ir $\phi$ yra atsitiktiniai
    elementai su reikšmėmis erdvėje $H^o_\alpha(\mathbb{H}).$ Tarkime yra tenkinamos
    sąlygos.
\begin{itemize}
    \item[i)] Kiekvienam diadiniam $\bm{t}\in [0,1]^d$, atsitiktinių
	$\mathbb{H}$-elementų apibendrinta seka $\phi_{\bm{n}}(t)$ yra
	asimptotiškai tiršta erdvėje $\mathbb{H}$.
   \item[ii)]  Kiekvienam $\eps>0$
    \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}}(\phi_{\bm{n}})|>\eps) = 0.
    \end{align*}
\end{itemize}
Tada apibendrinta seka $\phi_{\bm{n}}$ yra asimptotiškai tiršta erdvėje $\ab{H}^o(\mathbb{H})$.
\end{teorema}
\end{frame}



\begin{frame}
    \frametitle{Vynerio paklodė} 
    Vynerio procesas su reikšmėmis Hilberto erdvėje $\mathbb{H}$ ir kovariacijos operatoriumi
    $S$ yra Gausinis nulinio vidurkio $\mathbb{H}$ procesas tenkinantis
    \begin{align*}
	E\langle W(\bm{t}),x\rangle\langle W(\bm{s}),y\rangle =(t_1\wedge
	s_1)\dots(t_d\wedge s_d)\langle Sx,y\rangle 
    \end{align*}
    kiekvienam $\bm{t},\bm{s}\in [0,1]^d$ ir $x,y\in \mathbb{H}$.

    Remiantis Račkausko ir Suquet rezultatais $W(\bm{t})$ turi versiją
    $H^o_\alpha(\mathbb{H})$.

\end{frame}

\begin{frame}
    \frametitle{Pagrindinis rezultatas} 
     \begin{teorema}
	Tegu $X_{\bm{j}}$, $\bm{j}\le \bm{n}$ yra vienodai pasisikirstę
	atsitiktiniai dydžiai su reikšmėmis erdvėje $\mathbb{H}$.
	\begin{align*}
	(n_1\dots n_d)^{-1/2}\xi_{\bm{n}} \xrightarrow[m(\bm{n}) \to
	\infty]{\mathcal{D}} W \text{ erdvėje } \ab{H}^o(\mathbb{H}),
   \end{align*}
    
    %\vspace*{15pt}
    
    tada ir tik tada, kai
    
    %\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{ kai } m(\bm{n}) \to \infty.
    \end{align*},
    čia $m(\bm{n})=n_1\wedge\dots\wedge n_d$.
    
   \end{teorema}
\end{frame}
\begin{frame}
    \frametitle{Fiksuotos trajektorijos} 
    \begin{itemize}
	\item Fiksuotai trajektorijai $\bm{n}=(n_1,\dots,n_d)$ gauname sąlygą:
	    \begin{align*}
		\lim_{t\to\infty}t^{\frac{d}{d/2-\alpha}}P(\|{X_{\bm{1}}}\|\ge t)=0,
	    \end{align*}
	\item Trajektorijai $n_1^{1/p}=(n_2\dots n_d)^{1/2}$ gauname sąlygą:
	    \begin{align*}
		\lim_{t\to\infty}t^{1+p/2}P(\|{X_{\bm{1}}}\|\ge t)=0,
	    \end{align*}
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Vienareikšmė formuluotė} 
    Sąlyga
    \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{ kai } m(\bm{n}) \to \infty.
    \end{align*}
    yra ekvivalenti sąlygai
    \begin{align*}
	\sup_{t>0} t^pP(\|X_{\bm{1}}\|>t)<\infty
    \end{align*}
    kai $d>1$.
\end{frame}
\begin{frame}
    \frametitle{Palyginimas} 
    $p=1/(1/2-\alpha)$
    \begin{itemize}
	\item<1-4,6-> Ericksono sąlyga:  $E|X_1|^q<\infty$,   when $q>d/(1/2-\alpha)$.
	\item<2-3,5-> Račkausko ir Suquet sąlyga ($d=1$): $\lim t^p\mathbf{P}(|X_{\bm{1}}|>t)=0$
	\item<3-> Ekvivalenčios sąlygos
		    \begin{align*}
			\lim_{m(\bm{n})\to\infty}n_1\dots
			n_dP(\|X_{\bm{1}}\|>n_1^{1/p}n_2^{1/2}\dots
			n_d^{1/2})=0
		    \end{align*}
		    ir
		    \begin{align*}
			\sup_{t>0}t^pP(\|X_{\bm{1}}\|>t)<\infty \quad (d>1)	
		    \end{align*}
    \end{itemize}
    \begin{enumerate}
	\item<4,6-> Reikia $d$ „mažiau“ momentų.
	\item<5-> Užtenka silpno momento.
    \end{enumerate}
\end{frame}
\end{document}
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%\subsection{Trajektorijų erdvės}
\begin{frame}
    \frametitle{Trajektorijų erdvės} 
    Bendras atvejis:
    \begin{itemize}
	    \item $I^d=[0,1]^d$ with norm $|t|=\max\{|t_1|,\dots,|t_d|\}$,
		$t\in I^d$.
	    \item $\mathcal{A}$ collection of Borel subsets of $I^d$ with
		semimetric $\delta(A,B):=\sqrt{|A\Delta B|}$
	    \item $\mathcal{I}_d:=\{[0,t_1]\times\dots\times[0,t_d]\}$ with
		semimetric $\delta(A,B)$.
	\end{itemize}
    Range normed vector space:
	\begin{itemize}
	    \item $\mathbb{R}$ with norm $|t|$.
	    \item Hilbert space $\mathbb{H}$ with inner product
		$\sprod{\cdot,\cdot}$ and norm $\|\cdot\|_H$.
	    \item Separable Banach space $B$ with norm $\|\cdot\|$
	\end{itemize}
\end{frame}

\begin{frame}
    \frametitle{Spaces of functions} 
     $(T,d)$ index space with semimetric $d$, $(B,\|\|)$ - vector range space.
     Consider functions $x:T\to B$.
    \begin{itemize}
	\item $C(T,B)$ -- space of continuous functions with norm
	    $$\|x\|_\infty:=\sup_{t\in T}\|x(t)\|$$
	\item If $T$ is completely bounded with respect to $d$,
	    $C(T,\mathbb{R})$ is separable.
	\end{itemize}
\end{frame}
\begin{frame}
    \frametitle{H\" older spaces} 
    \begin{itemize}
	\item $H_\rho(T,B)$ -- of continuous functions satisfying $w_{\rho}(x,1)<\infty$,
	    where
	    \begin{align*} %\label{f:walpha}
		w_\rho(x,\delta)=\sup_{t,s\in T,d(t,s)<\delta}
		\dfrac{\|x(t)-x(s)\|}{\rho(d(t,s))}.
	    \end{align*}
	 \item $H^o_\rho(T,B)$ subset of $H_\rho(T,B)$ of functions satisfying
	    \begin{align*}
		\lim_{\delta\to\infty}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{Separability} 
    \begin{itemize}
	\item $H_\rho(T,B)$ - nonseparable Banach space with norm $\|x\|_\rho:=\|x\|_\infty+w_\rho(x,1)$.
	\item For $T$ - totaly bounded, $H_\rho(T,\mathbb{R})$ is
	      $\sigma$-compact separable linear space with norm $\|x\|_f$, where
	      $\rho=o(f)$.

	\item For $\rho=h^\alpha\log^\beta(c/h)$,  $H_\rho^o([0,1]^d,B)$ is
	    separable Banach space with Schauder decomposition.
%	      $$H_\rho^o([0,1]^d,B):=\{x\in  H_\rho([0,1]^d,B):\lim_{\delta\to\infty}w_\rho(x,\delta)=0\}$$.
	    \end{itemize}
\end{frame}

\subsection{Functional central limit theorem}
\begin{frame}
    \frametitle{Functional central limit theorem}
  
    FCLT-$C(T,B)$ (FCLT-$H_\rho(T,B)$) holds if
    \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*}
	in space $C(T,B)$, ($H_\rho(T,B)$),   where $m(\bm{n}):=n_1\wedge\ldots\wedge n_d$. 
  
  %\begin{itemize}
%	\item $W_d(\bm{t})=0$, jei nors vienas $t_i=0$.
%	\vspace*{20pt}
%	\item $W_d(\bm{t})\ \sim\ N(0,t_1\cdots t_d)$,
%	    $\bm{t}=(t_1,\ldots,t_d)$.
 %   	\vspace*{20pt}
%	\item $\mathbf{E}W_d(\bm{t})W_d(\bm{s})=(t_1\wedge
%	    s_1)\cdots(t_d\wedge s_d)$.
 %   \end{itemize}
\end{frame}
\subsection{Wiener sheet}
\begin{frame}
    \frametitle{Brownian motion}
    Index set $I=[0,1]$.
    \begin{itemize}
	\item $\mathbb{R}$-valued. $W(t)$ is zero mean Gausian with variance
	    $t$.
	\item $B$-valued with parameter $\mu$ (distribution of a Gaussian random
	    element $Y$ on $B$), - Gausian process with
	    independent increments such that $W(t)-W(s)$ has the same
	    distribution as $|t-s|^{1/2}Y$.
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Wiener sheet} 
    Index sets $\mathcal{A}$ and $I^d$.
    \begin{itemize}
	\item $W(A)$ -- Gaussian with $EW(A)=0$ and $EW(A)W(B)=|A\cap B|$.
	\item $\mathbb{H}$-valued, zero mean Gaussian process satisfying
	    \begin{align*}
		E\sprod{ W(\bm{t}),x}\sprod{W(\bm{s}),y}:=(t_1\wedge
		s_1)\times\dots\times (t_d\wedge s_d)\sprod{x,y}
	    \end{align*}
	    where $S$-- covariance operator in $\mathbb{H}$.
    \end{itemize}
\end{frame}
\section{Known results}
\subsection{Continuous function spaces}
\begin{frame}
    \frametitle{Continuous function spaces}
     \begin{itemize}
	 \item FCLT-$C([0,1],\mathbb{R})$. Donsker-Prokhorov theorem.
	    Necessary and sufficient condition $EX_1^2<\infty$.
	%    \vspace*{20pt}
	\item FCLT-$C([0,1],B)$. Kuelbs(1973), necessary and sufficient
	    condition $X_1\in CLT(B)$.
	\item FCLT-$C(\mathcal{A},\mathbb{R})$.  Bass (1985), Pyke (1986),
	    sufficient conditions $EX_1^2<\infty$ and
	    \begin{align*}
		\int_0^1\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}
\subsection{H\"older spaces} 
\begin{frame}
    \frametitle{One-dimensional index space} 
    \begin{itemize}
	\item FCLT-$H_\rho^o([0,1],\mathbb{R})$, $\rho=h^{\alpha}$.   Lamperti
	    (1962), sufficient condition: 
	    $$\mathbf{E}|X_1|^p<\infty,$$
	    where $p>1/(1/2-\alpha)$ and  $0<\alpha<1/2$.
	%\vspace*{20pt}
	\item FCLT-$H_\rho^o([0,1],B])$, $\rho=h^\alpha L(1/h)$, $0<\alpha\le
	    0$.  Ra\v ckauskas and Suquet (2004), necessary and sufficient
	    conditions:
	  %  \begin{salyga}
		$X_1\in CLT(B)$, and for each $A>0$	
		\begin{align*}
		\lim_{t\to\infty}tP(\|X\|_B\ge At^{1/2}\rho(1/t))=0
	    \end{align*}
	%\end{salyga}
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Two-dimensional index space}
    \begin{itemize}
	\item FCLT-$H_\alpha([0,1]^2,\mathbb{R})$. Ra\v ckauskas and Zemlys
	    (2005),  necessary and sufficient condition:
	    \begin{salyga}
	    \begin{align*}
		t_1t_2\mathbf{P}(|X_1|\ge t_1^{1/p}t_2^{1/2})\to 0, \text{ when }
		t_1\wedge t_2\to\infty, \text{ with } p=\frac{1}{1/2-\alpha}.
	    \end{align*}
	\end{salyga}
    \end{itemize}
\end{frame}
\begin{frame}
    \frametitle{Multi-dimensional index space}
    %\vspace*{40pt}
    \begin{itemize}
	\item Erickson (1981). Let $E|X_1|^p<\infty$, $r_1(\mathcal{A})\le r$,
	    and $q:=p/2-r>0$. 
	    \begin{salyga}	    
	    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 $\beta>1$.
	\end{salyga}
	 \item For $\mathcal{A}:=\mathcal{I}^d$, sufficient condition translates
	     to $E|X|^q<\infty$, for $q>\frac{d}{1/2-\alpha}$, $0<\alpha<1/2$.
     \end{itemize}
\end{frame}

%\section{Known results}
%\subsection{One dimensional case}

%\subsection{Two dimensional and multi-dimensional case}
%\begin{frame}
%    \frametitle{Summation process}
%    \begin{align*}
%	\xi_{\bm{n}}(\bm{t})=& S_{([n_1t_1],\ [n_2t_2])}+\\
%	&+(n_1t_1-[n_1t_1])S^{(2)}_{([n_1t_1]+1,[n_2t_2])}\\
%	&+(n_2t_2-[n_2t_2])S^{(1)}_{([n_1t_1], [n_2t_2]+1)}\\
%	&+(n_1t_1-[n_1t_1])(n_2t_2-[n_2t_2])X_{([n_1t_1]+1,  [n_2t_2]+1)},
%    \end{align*}
%    where
%   \begin{align*}
%    S_{(k,\ m)}=\sum_{j=1}^k\sum_{j=1}^m X_{(i,\ j)},\ \ S^{(1)}_{(k,\
%	m)}=\sum_{j=1}^m X_{(k, \  j)},\ \  S^{(2)}_{(k,\
%	m)}=\sum_{i=1}^k X_{(i,\ m)}.
%    \end{align*}
%\end{frame}
\section{Main result}
%\subsection{Formulation and comparison}
\begin{frame}
    %\vspace*{20pt}
    \frametitle{Extension to $H_\alpha([0,1]^d,\mathbb{H})$}
    
    \begin{teorema}
	Let $X_{\bm{j}}$, $\bm{j}\le \bm{n} $ i.i.d. zero mean $H$-valued random
	variables
	\begin{align*}
	(n_1\dots n_d)^{-1/2}\xi_{\bm{n}} \xrightarrow[m(\bm{n}) \to
	\infty]{\mathcal{D}} W \text{ in the space } \ab{H}^o(\mathbb{H}),
   \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*}
   \end{teorema}
\end{frame}

\begin{frame}
    \frametitle{Fixed trajectory} 
    Let $\bm{n}=(n,\dots,n)\in \mathbb{N}^d$
    \begin{teorema}
	Let $X_{\bm{j}}$, $\bm{j}\le \bm{n} $ i.i.d. zero mean $H$-valued random
	variables
	\begin{align*}
	n^{-d/2}\xi_{n} \xrightarrow[n \to
	\infty]{\mathcal{D}} W \text{ in the space } \ab{H}^o(\mathbb{H}),
   \end{align*}
    
    %\vspace*{15pt}
    
    if and only if
    
    %\vspace*{5pt}
   
   \begin{align*}
	\lim t^{\frac{d}{d/2-\alpha}}\mathbf{P}\bigl(\|X_{\bm{1}}\|>t\bigr)=0.
	   \end{align*}
       \end{teorema}
\end{frame}

\begin{frame}
    \frametitle{Comparison of results}
    Old condition:  $\mathbf{E}|X_1|^q<\infty$,   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 t_d^{1/2}\bigr)  \to
    0, \text{ when } m(\bm{n}) \to \infty.$ 
    \vspace*{20pt}

    \begin{itemize}
	\item For $d$ dimensional case Erickson's result is improved. You ``need''
	    $d$ times less moments.
	    \begin{align*}
		\lim_{t\to\infty}t^p\mathbf{P}(|X_{\bm{1}}|\ge t)=0.
	    \end{align*}
	\item When path is fixed: $n_1=\dots=n_d=n$, you only need less moments
	    \begin{align*}
		\lim_{t\to\infty}t^{\frac{d}{d/2-\alpha}}\mathbf{P}(|X_{\bm{1}}|\ge t)=0.
	    \end{align*}
%	    $\frac{d}{1/p+(d-1)/2}=\frac{2dp}{2+p(d-1)}<\frac{2d}{d-1}$
    \end{itemize}
\end{frame}

%\subsection{Details of the proof}
\begin{frame}{Convergence criteria}
    %\vspace*{20pt}
    \begin{itemize}
	\item $(\xi_{\bm{n}}, \bm{n}\in\mathbb{N}^d)$ is a net with directed set
	    $\mathbb{N}^d$ and relation $\bm{j}\le \bm{n}\Leftrightarrow j_i\le n_i$,
	    $i=1,\ldots,d$.
	    %\vspace*{20pt}
	\item Prokhorov theorem. Asymptoticaly tight sequence of randome
	    elements is relatively compact.
	\item  Convergence of finite-dimensional distributions.
	\item The characterisation of compact in space
	    $\ab{H}^o(\mathbb{H})$.
    \end{itemize}
\end{frame}

\begin{frame}    
Let $\{\zeta_{\bm{n}}, \bm{n}\in\mathbb{N}^d\}$ be a net of random elements from
$\ab{H}^o(\mathbb{H})$ satisfying conditions:
\begin{enumerate}
    %\item Baigtiniamačiai $\zeta_{\bm{n}}$ skirstiniai silpnai konverguoja į
%	kokio nors erdvės $\ab{H}^o\bigl([0,1]^d\bigr)$ atsitiktinio elemento $\zeta$
%	baigtiniamačius skirstinius, kai  $n_1\wedge\dots\wedge n_d\to\infty$.
\item For every $\bm{t}\in [0,1]^d$, $\zeta_{\bm{n}}$ is asymptoticaly tight.
	%$\lim_{a\to \infty}\sup_{\bm{n}} P(\sup_{\bm{t}\in
        %T}|\zeta_{\bm{n}}(\bm{t})|>a)=0$
    \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}
Then $\zeta_{\bm{n}}$ is tight.
\end{frame}
\end{document}

