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\title{Funkcin� ribin� teorema dvima�io indekso sumavimo procesams}
\author{Alfredas Ra�kauskas,\\ Vaidotas Zemlys}
\institution{Vilniaus Universitetas, Matematikos ir Informatikos Fakultetas,
Ekonometrin�s analiz�s katedr
%\email{Vaidotas.Zemlys@maf.vu.lt}

\begin{document}
\maketitle
\begin{slide}{Darbo esm�}
    \vspace*{40pt}
    \begin{itemize}
	\item   Papildytas Erickson rezultatas  
	    \vspace*{20pt}
	\item   Apibendrintas Ra�kausko ir Suquet rezultatas

	\end{itemize}
\end{slide}

\begin{slide}{Sumavimo procesas}
    \vspace*{20pt}
    \begin{itemize}
	\item ($X_{\bm{j}}$, $\bm{j}\in \mathbb{N}^d$) -- v. p. r. a. d. su
	    nuliniu vidurkiu. 
	    \vspace*{10pt}
	\item $\bm{n}=(n_1,\ldots,n_d)$, $\bm{t}=(t_1,\ldots,t_d)\in [0,1]^d$ 
	    \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
	    [0,t_1]\times\cdots\times[0,t_d]|X_{\bm{j}},
	    \end{align*}
	    �ia
	    \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{slide}

\overlays{2}{%
\begin{slide}{Vienmatis atvejis}
    \onlySlide*{1}{
    \vspace*{40pt}
    �prastinis lau��i� procesas
    \begin{align*}
	\xi_n(t)=S_{[nt]}+(nt-[nt])X_{[nt]+1},\quad t\in[0,1],
    \end{align*}
    �ia 
    \begin{align*}
	S_k=X_1+X_2+\ldots+X_k, \quad k=1,2,\ldots
    \end{align*}
    }%
    \onlySlide*{2}{\includegraphics{W10.ps}}
\end{slide}}

\begin{slide}{Donskerio-Prochorovo teorema}
	    \vspace*{10pt}
	\begin{align*}
		n^{-1/2}\xi_n\xrightarrow{\mathcal{D}}W \text{ erdv�je } C[0,1] \Leftrightarrow \mathbf{E}X_1^2<\infty,
    \end{align*}
	 \vspace*{10pt}
	 �ia $W(t)$ - Vynerio procesas arba Brauno judesys.
	\vspace*{10pt}
	\begin{itemize}
		\item $W_0=0$.
     	\vspace*{10pt}
     	\item $W_t\ \sim\ N(0,t)$.
     	\vspace*{10pt}
     	\item $\mathbf{E}W_tW_s=t\wedge s$.
     	\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*}
    %	nepriklausomi atsiktiniai dyd�iai.
    	\end{itemize}
\end{slide}

\begin{slide}{Hiolderio erdv�s}
    \vspace*{20pt}
       \begin{itemize}
	   \item Tegu $0<\alpha\le 1$, $\ab{H}^o\bigl([0,1]^d\bigr)$ yra aib� toki� reali� tolyd�i�
	funkcij� $x:[0,1]^d\to \mathbb{R}$, kurioms $\lim_{\delta\to   0}\wab{x}=0$, �ia
	\begin{align*} %\label{f:walpha}
	    \wab{x}=\sup_{\bm{t},\bm{s}\in[0,1]^d,0<|\bm{t}-\bm{s}|<\delta}
	    \dfrac{|x(\bm{t})-x(\bm{s})|}{|\bm{t}-\bm{s}|^\alpha}.
        \end{align*}
	\item Separabili Banacho erdv� su norma
	    \begin{align*}
	    \normab{x}=|x(0)|+\wab[1]{x}.
	    \end{align*}
       \end{itemize}
\end{slide}

\begin{slide}{D-P apibendrinimas}
	\begin{align*}
		n^{-1/2}\xi_n\xrightarrow[n\to\infty]{\mathcal{D}}W, \text{ erdv�je } \ab{H}^o([0,1])
    \end{align*}
	\begin{itemize}
		\item Lamperti (1962) �rod� pakankam� s�lyg�: 
		\begin{align*}
			\mathbf{E}|X_1|^p<\infty,
		\end{align*}
		�ia $p>1/(1/2-\alpha)$ ir  $0<\alpha<1/2$.
		\item Ra�kauskas ir Suquet (2001) papild� Lamperti rezultat� �rodydami
	    b�tin� ir pakankam� s�lyg�:
	    \begin{align*}
		\lim_{t\to\infty}t^p\mathbf{P}(|X_1|\ge t)=0 \text{ su } p=\frac{1}{1/2-\alpha}.
	    \end{align*}
	\end{itemize}
\end{slide}

\overlays{3}{%
\begin{slide}{Dvimatis atvejis}
    \onlySlide*{1}{
	\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*}
    �ia
   \begin{align*}
    S_{(k,\ m)}=\sum_{i=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*}}%
	\onlySlide*{2}{\includegraphics{net.ps}}%
	\onlySlide*{3}{\includegraphics{W10x10.ps}}%
\end{slide}}

\begin{slide}{Ribin�s teoremos}
\vspace*{20pt}
Tolyd�ius sumavimo procesus dvima�iu atveju nagrin�jo Kuelbs (1968) . Jis
apsiribojo indeksu $\bm{n}=(n,n)$ ir �rod�, kad
\begin{align*}
    \xi_{(n,n)}\xrightarrow[n\to\infty]{\mathcal{D}}W_2,
\end{align*}
erdv�je $C([0,1]^2)$, jei $E|X_{(1,1)}|^{3+\delta}<\infty$, kokiam nors
$\delta>0$. �ia $W_2$ -- dvimat� Vynerio paklod�.
\end{slide}

\begin{slide}{Vynerio paklod�}
    \vspace*{30pt}
    \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{slide}

\overlays{2}{%
\begin{slide}{Vynerio paklod�s savyb�s}
    \onlySlide*{1}{
    \vspace*{30pt}
    \begin{itemize}
	\item Fiksavus vien� $t_i$, gaunama Vynerio paklod� $W_{d-1}$.
	\vspace*{20pt}
	\item Nesikertan�iuose sta�iakampiuose, nepriklausomi
	    atsitiktiniai dyd�iai. $\mathbf{E}W(\bm{s})(W(\bm{t})-W(\bm{s}))=0$, $\bm{s}\le\bm{t}.$
	\vspace*{20pt}
	\item Trajektorijos priklauso erdvei $\ab{H}^o\bigl([0,1]^d\bigr)$, $0<\alpha<1/2$.
    \end{itemize}}%
    \onlySlide*{2}{\includegraphics{Wnet.ps}}% 
\end{slide}}

\begin{slide}{Daugiamatis atvejis}
	\vspace*{10pt}
 	I� Erickson (1981) rezultat� i�plaukia, kad erdv�je $\ab{H}^o\bigl([0,1]^d\bigr)$
	\begin{align*}
	    	(n_1\cdots n_d)^{-1/2}\xi_n\xrightarrow{\mathcal{D}}W_d,
		\text{ kai } \bm{n}\to\infty,
	\end{align*}
	jei 
	\begin{align*}
		\mathbf{E}|X_1|^q<\infty, \text{ kai }  q>d/(1/2-\alpha).
	\end{align*}

�ia $\bm{n}\to\infty$ rei�kia, kad $n_1\wedge\ldots\wedge n_d\to\infty$.
\end{slide}

\begin{slide}{Pagrindinis rezultatas}
    \vspace*{20pt}
    \begin{align*}
	(n_1n_2)^{-1/2}\xi_{\bm{n}} \xrightarrow[n_1\wedge n_2 \to
	\infty]{\mathcal{D}} W_2 \text{ erdv�je } \ab{H}^o\bigl([0,1]^2\bigr),
    \end{align*}
    
    \vspace*{15pt}
    
    tada ir tik tada, jei
    
    \vspace*{5pt}
   
    \begin{align*}
	    t_1t_2\mathbf{P}\bigl(|X_{\bm{1}}|>t_1^{1/p}t_2^{1/2}\bigr)\to 0, \text{ kai } t_1\wedge t_2 \to \infty,
	   \end{align*}
	   �ia $p=1/(1/2-\alpha)$.
\end{slide}

\begin{slide}{Konvergavimo kriterijus}
    Tegu $\{\zeta_{\bm{n}}, \bm{n}\in\mathbb{N}^2\}$ yra atsitiktini� erdv�s 
    $\ab{H}^o\bigl([0,1]^2\bigr)$ element� apibendrinta seka tenkinanti �ias s�lygas:

\begin{enumerate}
    \item Baigtiniama�iai $\zeta_{\bm{n}}$ skirstiniai silpnai konverguoja �
	kokio nors erdv�s $\ab{H}^o\bigl([0,1]^d\bigr)$ atsitiktinio elemento $\zeta$
	baigtiniama�ius skirstinius, kai  $n_1\wedge n_2\to\infty$.
    \item $\lim_{a\to \infty}\sup_{\bm{n}} P(\sup_{\bm{t}\in
        T}|\zeta_{\bm{n}}(\bm{t})|>a)=0$
    \item  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}}(\zeta_{\bm{n}})|>\eps) = 0.
        \end{align*}
\end{enumerate}
\end{slide}
\begin{slide}{Apibr��imai}
	Kiekvienai funkcijai $x\in \ab{H}^o$:
\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*}
�ia $V_j=W_j\backslash W_{j-1}$, $W_j=\{k2^{-j};0\le k\le 2^j\}^2$.
\begin{align*}
  v_i^\pm=
  \begin{cases}
    v_i\pm2^j, & \text{kai}\; k_i\; \text{yra nelyginis};  \\
     v_i,& \text{kai}\; k_i\; \text{yra lyginis}.
  \end{cases}
\end{align*}    
\end{slide}
\begin{slide}{Prochorovo teorema}
    \vspace*{20pt}
	Tegu $(X_\alpha,\alpha\in I)$ yra  apibendrinta separabilios erdv�s atsitiktini�
element� seka. Jei $(X_\alpha)$ yra asimptoti�kai tir�ta, t.y. kiekvienam
$\eps>0$ egzistuoja toks kompaktas $K_\eps$, kad
\begin{align*}
    \liminf_{\alpha}P(X_\alpha \in K_\eps)>1-\eps.
\end{align*}
tai apibendrinta seka $(X_\alpha)$ yra reliatyviai kompakti�ka, (1.3.9 teorema 21
psl. van der Vaart ir Wellner (1996))
\end{slide}


\begin{slide}{Arcela-Askoli teorema}
	\vspace*{20pt}
	Erdv�s $\ab{H}^o$ poaibis $A$ yra reliatyviai kompakti�ka aib� tada ir tik
tada, kai tenkinamos s�lygos
\begin{align*}
  &\sup_{x\in A}\sup_{t\in T}|x(t)|<\infty  \\%label{AApirma:salyga}\\
  \intertext{ir}
  &\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{slide}

\begin{slide}{�inomi faktai}
\vspace*{20pt}
\begin{itemize}
	\item $(\xi_{\bm{n}}, \bm{n}\in\mathbb{N}^2)$ apibendrinta seka su
	    s�ry�iu $\bm{j}\le \bm{n}\Leftrightarrow j_i\le n_i$,
	    $i=1,\ldots,d$.
	\vspace*{20pt}
	\item Proceso $(n_1n_2)^{-1/2}\xi_{\bm{n}}(\bm{t})$ baigtiniama�iai pasiskirstymai konverguoja � Vynerio proceso baigtiniama�ius
	 pasiskirstymus, kai $n_1\wedge n_2\to \infty$. (Erickson (1981))
\end{itemize}

\end{slide}

\begin{slide}{B�tinumas}
	Turime 
\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*}
tod�l	
\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{slide}

\begin{slide}{Pakankamumas}
	\vspace*{20pt}
	D�l simetrijos u�tenka patikrinti, kad kiekvienam $\eps>0$ galioja
	\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*}
�ia $t_k=k2^{-j}$ and $s_\ell=\ell 2^{-j}$.
	
\end{slide}

\begin{slide}{Pakankamumas}
	Skirtumui $\Delta_{\bm{n}}(t, t'; s)=|\xi_{(n_1, n_2)}(t', s)-\xi_{(n_1,n_2)}(t, s)|$, teisinga:
	
\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*}
�ia
\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{slide}

\begin{slide}{Rezultat� palyginimas}
    Sena s�lyga:  $\mathbf{E}|X_1|^q<\infty$,   kai $q>2/(1/2-\alpha)$.
    
    Nauja s�lyga: $ t_1t_2\mathbf{P}\bigl(|X_{\bm{1}}|>t_1^{1/p}t_2^{1/2}\bigr)  \to 0, \text{ kai } t_1\wedge t_2 \to \infty.$ 
    \vspace*{20pt}

    \begin{itemize}
	\item Pagerintas Erickson rezultatas dvima�iam atvejui. ``Reikia''
	    dvigubai ma�iau moment�.
	    \begin{align*}
		\lim_{t\to\infty}t^p\mathbf{P}(|X_{\bm{1}}|\ge t)=0.
	    \end{align*}
	\item Einant fiksuotais keliais: $n_2=O(n_1^r)$, $r\ge 1$ u�tenka ma�iau moment�, negu vienma�iu
	    atveju.
	    \begin{align*}
		\lim_{t\to\infty}t^{\frac{r+1}{r/p+1/2}}\mathbf{P}(|X_{\bm{1}}|\ge t)=0.
	    \end{align*}
    \end{itemize}

\end{slide}

\overlays{4}{%
\begin{slide}{Skirtingas moment� skai�ius} 
    \onlySlide*{1}{\includegraphics{W60x601.ps}}% 
    \onlySlide*{2}{\includegraphics{W60x602.ps}}% 
    \onlySlide*{3}{\includegraphics{W60x603.ps}}% 
    \onlySlide*{4}{\includegraphics{W60x60r.ps}}%
    \onlyInPS{\includegraphics{W60x60r.eps}}% 
\end{slide}}

\begin{slide}{Nei�spr�stos problemos}
    \vspace*{20pt}
    \begin{itemize}
        \item Hiolderio erdv�s su bendresniu tolydumo moduliu $\rho$.
		\item	Rasti b�tinas ir pakankamas s�lygas, kai $d\ge 2$.
	    \vspace*{10pt}   
	\item	Rasti b�tinas ir pakankamas s�lygas atvejui su Vynerio
	    paklode $\{W(A),A\in T_{\mathcal{B}}\}$, �ia $T_{\mathcal{B}}$
	    -- vis� $[0,1]^d$ Borelio poaibi�, kuri� Lebego matas baigtinis,
	    aib�.
	    \vspace*{10pt}
	\item Save normuojan�io apibendrinto lau��i� proceso konvergavimas.
    \end{itemize}
\end{slide}

\begin{slide}{Papildoma informacija}
    \vspace*{20pt}
    \begin{itemize}
	\item Skaidr� daryta su \LaTeX \ naudojant \textbf{prosper} paket� i�
	    \href{http://prosper.sourceforge.net/}{{\blue
	    http://prosper.sourceforge.net}} 
    \vspace*{10pt}
	\item Grafikai daryti naudojant statistin� paket� \textbf{R} i�
	     \href{http://www.r-project.org/}{{\blue
	    http://www.r-project.org}} 
    \vspace*{10pt}
    \item Skaidres, \LaTeX\ dokument� ir \textbf{R} kod� galima rasti
	     \href{http://mif.vu.lt/~zemlys/mokslas/}{{\blue
	    http://mif.vu.lt/$\sim$zemlys/mokslas/}} 
    \end{itemize}
\end{slide}
\end{document}
