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Z toho plyne, že takto definovaná kovarianční matice je symetrická. +Všimneme si, že platí $\Cov(X, X) = \Var X$ a $\Cov(X, Y) = \Cov(Y, X)$. Z toho plyne, že takto definovaná kovarianční matice je symetrická a navíc $\Var \vec{X} \left[Cov(X_i, X_j)\right]_{i,j=1\dots d}$. \hfill \textit{konec 9. přednášky (17.3.2025)} + +Budeme pokračovat základními vlastnostmi variančních matic, které se chovají podobně rozptylu jednorozměrné náhodné veličiny. + +\begin{theorem}[Vlasnosti varianční matice] + Máme-li náhodný vektor $\vec{X}$ a reálné vektory $\vec{a}, \vec{b}$ takové, že následující výrazy mají smysl, potom + $$ \E (\vec{a}^T\vec{X} + \vec{b}) = \vec{a}^T\E \vec{X} + \vec{b}, \Var(\vec{a}^T\vec{X} + \vec{b}) = \vec{a}^T (\Var \vec{X}) \vec{a}.$$ + + Máme-li náhodný vektor $\vec{X}$ a $\vec{A}, \vec{B}$ jsou reálné matice, pak + $$ \E(A\vec{X} + B) = A\E \vec{X} + B, \Var(A\vec{X} + B) = A(\Var\vec{X})A^T. $$ +\end{theorem} + +\begin{proof} + Z definice násobení matic a linearity operátoru $\E$. +\end{proof} + +\begin{definition} + Pro danou náhodnou veličinu $X$ definujeme \textit{momentovou vytvořující funkci} (MGF) vztahem + $$ \psi_X(t) = \E[\exp(tX)] = \int_\R e^{tx} dP_X(x) $$ + pro $t \in \R$, pokud pravá strana existuje. Speciální případ $\psi_X(-t)$ nazýváme \textit{Laplaceovou transformací} $X$. +\end{definition} + +\begin{theorem}[Vlastnosti MGF] + Platí následující vlastnosti MGF: + \begin{enumerate}[(i)] + \item Existuje-li $\varepsilon > 0$ takové, že na $(-\varepsilon, \varepsilon)$ existuje $\psi_X(t)$, potom $\psi_X^{(m)}(0) = \E X^m, m \in \N_0$; + \item Pokud $Y = aX + b$, pak $\psi_Y(t) = e^{bt}\psi_X(at)$; + \item Pokud $X_1, \dots, X_d$ jsou nezávislé a $Y = \sum_{l = 1}^d X_l$, pak platí $\psi_Y(t) = \prod_{l = 1}^d \psi_{X_l}(t).$ + \end{enumerate} +\end{theorem} + +\begin{proof} + Dokážeme první vlastnost. Případ $m = 0$ je triviální, nechť tedy máme $m > 0$. Nejdříve budeme uvažovat případ $m = 1$ a chceme použít větu o konvergentní majorantě. Nechť tedy + $$ g(X) := e^\frac{-\varepsilon x}{2} + e^\frac{\varepsilon x}{2}, $$ + potom platí $\exp{tx} \leq g(x)$ pro všechna $t \in [-\varepsilon/2, \varepsilon/2]$ a libovolné $x \in \R$. Dále z předpokladu máme, že + $$ \int_\R g(x) dP_X(x) = \psi_X(-\varepsilon/2) + \psi_X(\varepsilon/2) < +\infty. $$ + Dostáváme, že $g$ je hledaná konvergentní majoranta. Z věty o konvergentní majoranty tedy můžeme provést záměnu integrálu a derivace. + $$ \odv*{\psi(t)}{t} = \odv{}{t} \int_\R e^{tx} dP_X(x) = \int_\R xe^{tx} dP_X(x) \overset{t = 0}{=} \int_\R xdP_X(x) = \E X^1.$$ + Zbytek se dokáže indukcí s použitím stejné majoranty, v $m$-tém kroku dostaneme $\int_\R x^mdP_X(x) =: \E X^m$. + + Druhou vlastnost dokážeme přímým rozepsáním definice + $$ \psi_Y(t) = \psi_{aX + b}(t) = \E[\exp(taX + tb)] = \E[\exp\{atX\}e^{tb}] = $$ + $$ e^{tb}\E[\exp(atX)] = e^{tb}\psi_X(at). $$ + + Nakonec, poslední vlastnost se dokáže následně + $$ \psi_Y(t) = \psi_{\sum_{l=1}^d X_l} (t) = \E[\exp\{t \sum_{l = 1}^d] = \E\left[\prod_{l = 1}^d e^{tX_l}\right]. $$ + Dále využijeme nezávislost (která se přenáší i na veličiny transformované stejnou měřitelnou funkcí) a dostaneme + $$ \E\left[\prod_{l = 1}^d e^{tX_l}\right] = \prod_{l = 1}^d \E(e^{tX_l}) = \prod_{l = 1}^d \psi_{X_l}(t). $$ +\end{proof} + +Poznámka: pokud $\psi_X(t) = \psi_Y(t)$ pro všechna $t$ v nějakém otevřeném intervalu kolem $0$, pak $X$ a $Y$ se rovnají v distribuci. + +\begin{definition} + Pro danou náhodnou veličinu $X$ definujeme \textit{charakteristickou funkci} (CF) vztahem + $$ \varphi_X(t) = \E[\exp(itX)] = \int_\R e^{itx} dP_X(x) $$ + pro $t \in \R$. +\end{definition} + +Na rozdíl od momentové vytvořující funkce takto definovaná charakteristická funkce je dobře definovaná pro všechna $t \in \R$. Opět máme speciální název pro vyhodnocení charakteristické funkce v bodě $-t$, říkáme tomu \textit{Fourierova transformace}. Z definice exponenciály z komplexní analýzy okamžitě dostáváme vyjádření $\phi_X(t) = \E\cos(tX) + i\E\sin(tX)$. + +\begin{theorem}[Vlastnosti CF] + Platí následující vlastnosti CF: + \begin{enumerate}[(i)] + \item $\varphi_X$ existuje pro jakékoli rozdělení $X$; + \item $\varphi_X(0) = 1$; + \item $|\varphi_X(t)| \leq 1$ pro všechna $t \in \R$; + \item $\varphi_X$ je stejnoměrně spojitá, tedy $\forall \varepsilon > 0 \exists \delta > 0: |\varphi_X(t) - \varphi_X(s)| \leq \varepsilon$ kdykoli $|t - s| \leq \delta$; + \item $\varphi_{aX + b}(t) = e^{ibt} \varphi_{X}(at)$ pro všechna $t, a, b \in \R$; + \item $\varphi_{-X}(t) = \bar{\varphi}_X(t)$ (komplexně sdružená funkce); + \item $\varphi_X(t) \in \mathbb{R} \forall t \in \R$ právě tehdy když rozdělení je symetrické kolem bodu $t = 0$. + \item Jsou-li $X, Y$ nezávislé, potom $\varphi_{X + Y}(t) = \varphi_X(t)\varphi_Y(t)$. + \end{enumerate} +\end{theorem} + +\begin{proof} + Budeme dokazovat vlastnosti postupně + \begin{enumerate}[(i)] + \item Víme, žse pro všechna $x$ a všechna $t$ platí $|e^{itx}|^2 = \sin^2(tx) + \cos^2(tx) = 1$. Pak $\E |e^{itx}|^2 = 1$, z Jensenovy nerovnosti (bude později) máme, že $\E |e^{itx} | \leq \sqrt{\E |e^{itx}|^2} = 1$ a tedy $e^{itx}$ je integrovatelná. + \item Přímým dosazením dostaneme $\int_\R dP_X(x) = 1$. + \item Viz důkaz vlastnosti (i). + \item Položme $h := s - t$, potom + \begin{align*} + |\varphi_X(t) - \varphi_X(s)| = &|\E[e^{itX}] - \E[e^{i(t + h)X}]| \leq \\ + & \E [|e^{itX}(1 - e^{ihX}|] \leq \\ + & \E [|e^{itX}|\cdot|1 - e^{ihX}|] \leq \E[|e^{ihX} - 1|]. + \end{align*} + Víme, že $e^{ihX} - 1 \rightarrow 0$ když $h \rightarrow 0$ a zároveň $|e^{ihX} - 1| \leq 2$. Máme tedy konvergentní majorantu. Dle Lebesgueovy věty tedy platí $\lim_{h\rightarrow 0} \E |e^{ihX} - 1| = 0$. Z toho již plyne stejnoměrná spojitost. + \item Z definice dostáváme + $$ \varphi_{aX + b}(t) = \E e^{it(aX + b)} = e^{ibt} \E e^{itaX} = e^{ibt} \varphi_X(at).$$ + \item Využijeme přepisu do goniometrického tvaru (viz poznámka před touto větou), dostaneme + $$ \varphi_{-X}(t) = \E[\cos(-tX) + i\sin(-tX)] = \E[\cos(tX) -i\sin(tX)] = \bar{\varphi}_X(t).$$ + \item Nechť nejdříve $X$ má rozdělení symetrické kolem $0$, potom $X \overset{d}{=} -X$, z čehož máme $\varphi_{-X}(t) = \varphi_{X}(t)$. Aplikací již dokázané vlastnosti (v) máme, že $\varphi_{X} = \bar{\varphi}_X(t)$, tedy $\varphi_X(t) \in \R$. Opačná implikace se dokáže stejným postupem v opačném pořadí. + \item Rozepsání definice + $$ \varphi_{X + Y}(t) = \E[e^{it(X + Y)}] = \E[e^{itX}e^{itY}], $$ + dále díky nezávislosti dostáváme + $$ \E[e^{itX}e^{itY}] = \E[e^{itX}]\E[e^{itY}] = \varphi_X(t)\varphi_Y(t). $$ + \end{enumerate} +\end{proof} + +Následující věta nám umožňuje jednoznačně popisovat rozdělení jak podle distribuční funkce, tak i podle charakteristické funkce. + +\begin{theorem}[Leviho inverzní formule pro charakteristickou funkci] + Pro jakékoli rozdělení $X$ a libovolné $a < b$ platí + $$ \lim_{T \rightarrow \infty} \frac{1}{2\pi} \int_{-T}^T \frac{e^{-ita} - e^{-itb}}{it} \varphi_X(t) dt = P[a < X < b] + \frac{P[X = a] + P[X = b]}{2}.$$ +\end{theorem} + +\begin{proof} + Idea je taková, že postupně budeme rozdělovat integrály a odhadovat hodnoty těchto omezených integrálů. + Mějme $T \in \R$ a $a < b$, potom + $$ \frac{1}{2\pi} \int_{-T}^T \frac{e^{ita} - e^{itb}}{it} \varphi_X(t) dt = \frac{1}{2\pi} \int_{-T}^T \frac{e^{ita} - e^{itb}}{it} \int_\R e^{itx} dP_X = $$ + $$ \int_{-T}^T \int_{-\infty}^{\infty} \frac{e^{it(x - a)} - e^{it(x - b)}}{2\pi it} dP_Xdt \overset{Fubini}{=} \int_{-\infty}^\infty \int_{-T}^T \frac{e^{it(x - a)} - e^{it(x - b)}}{2\pi it} dtdP_X. $$ + Všimneme si, že pro každou konstantu $c \in \R$ platí $\int_{-T}^T \frac{e^{itc}}{2it}dt = \int_0^T \frac{\sin(tc)}{t} dt$. Potom platí + $$ \frac{1}{2\pi} \int_{-T}^T \frac{e^{ita} - e^{itb}}{it} \varphi_X(t) dt = $$ + $$ \int_{-\infty}^{\infty}\frac{1}{\pi} \left[\int_0^T \frac{\sin(t(x - a))}{t} dt - \int_0^T \frac{\sin(t(x - b))}{t} dt \right] dP_X.$$ + Když pošleme $T$ do nekonečna, dostaneme následující hodnoty + $$ \frac{1}{\pi} \int_0^T \frac{\sin(t(x - a))}{t} dt \overset{T\rightarrow \infty}\rightarrow + \begin{cases} + -\frac{1}{2}, x < a, \\ + \frac{1}{2}, x > a, \\ + 0, x = a. + \end{cases}$$ + Potom + $$ \frac{1}{\pi} \left[\int_0^T \frac{\sin(t(x - a))}{t} dt - \int_0^T \frac{\sin(t(x - b))}{t} dt \right] \overset{T \rightarrow \infty}\rightarrow \begin{cases} + \frac{1}{2}, x = a, b \\ + 1, a < x < b, + 0, \text{jinak}. + \end{cases}$$ + Dosazením do předchozího vzorce a užitím Lebesgueovy věty o konvergentní majorantě dostáváme + $$ \frac{1}{2\pi} \int_{-T}^T \frac{e^{ita} - e^{itb}}{it} \varphi_X(t) dt = P[a < X < b] + \frac{P[X = a] + P[X = b]}{2}. $$ +\end{proof} + +Z předchozí věty okamžitě plyne následující důsledek. + +\begin{corollary}[Jednoznačná charakterizace rozdělení] + Platí $\varphi_X = \varphi_Y \Leftrightarrow X \overset{d}{=} Y$. +\end{corollary} + +Nakonec definujeme charakteristickou funkci pro náhodné vektory. Obdobným způsobem pro ní můžeme dokázat vlastnosti, které platí pro jednorozměrné náhodné veličiny. + +\begin{definition} + \textit{Charakteristická funkce} náhodného vektoru $\vec{X} = [X_1, \dots, X_d]^T$ je definována vztahem + $$ \varphi_{\vec{X}}^{\vec{t}} = \E [e^{i\vec{t}^T\vec{X}}] = \int_\R e^{i\vec{t}^T\vec{X}} dP_{\vec{X}} $$ + pro $\vec{t} \in \R^d$. +\end{definition} + +\hfill \textit{konec 10. přednášky (18.3.2025)}