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Potom existuje $\varepsilon_0 > 0$ takové, že $|F_X(x - c) - F_X(x - c - \varepsilon)| < \frac{\eta}{3}$ pro každé $|\varepsilon| < \varepsilon_0$. + + Jelikož $F_X$ je distribuční funkce, má nejvýše spočetně mnoho bodů nespojitosti. Dokážeme tuto vlastnost. Nechť tedy $D$ je množina bodů nespojitosti $F_X$. Pak $\forall y \in D: F_X(y_-) < F_X(y^+)$, tedy existuje racionální číslo $q_y \in \mathbb{Q}$ takové, že $F_X(y_-) < q_y < F_X(y^+)$. Jelikož $F_X$ je neklesající, pak $y \neq z \in D \implies q_y \neq q_z$. Tedy zobrazení $y \mapsto q_y$ je prosté. + + Z toho máme, že v každém okolí bodu $x - c$ můžeme nalézt vlevo i vpravo od $x - c$ nějaký bod, v němž je $F_X$ spojitá. Z definice spojitosti existuje $0 < \varepsilon < \varepsilon_0$ takové, že $F_X$ je spojitá v $x - c + \varepsilon$ i v $x - c - \varepsilon$. Potom + $$ P(X_n + Y_n \leq x) = P(X_n + Y_n \leq x \cap |Y_n - c| < \varepsilon) + P(X_n + Y_n \leq x \cap |Y_n - c| \geq \varepsilon) \leq $$ + $$ P(X_n + c \leq x + \varepsilon \cap |Y_n - c| < \varepsilon) + P(|Y_n -c| \geq \varepsilon) \leq $$ + $$ P(X_n \leq x - c + \varepsilon) + P(|Y_n - c| \geq \varepsilon).$$ + + Jelikož $F_X$ je spojitá v bodě $x - c + \varepsilon$ a $Y_n \overset P \to c$, existuje $n_1 \in \N$ takové, že pro všechna $n \geq n_1$ platí $P(X_n \leq x - c + \varepsilon) \leq P(X_n \leq x - c) + \frac{\eta}{3}$ a zároveň $P(|Y_n - c| \geq \varepsilon) \leq \frac{\eta}{3}$. Dohromady pro $n \geq n_1$ máme + $$ P(X_n + Y_n \leq x) \leq P(X \leq x - c + \varepsilon) + \frac{2}{3}\eta \leq P(X \leq x - c) + \eta. $$ + Opačná nerovnost se dokáže analogicky. Pro $n \geq \max\{n_1, n_2\}$ dohromady máme + $$ P(X + c \leq x) - \eta \leq P(X_n + Y_n \leq x) \leq P(X + c \leq x) + \eta. $$ + Jelikož $\eta > 0$ bylo voleno libovolně, věta je dokázaná. +\end{proof} + +Z této věty okamžitě plyne následující důsledek (neboť konvergence v pravděpodobnosti implikuje konvergenci v distribuci, viz Věta \ref{thm-convergence-types}, vlastnosti (iv) a (v)). + +\begin{corollary} + Pokud $x_n \overset P \to a \in \R$ a $Y_n \overset P b \in \R$, potom $X_n + Y_n \overset P \to a + b$ a $X_nY_n \overset P \to ab$. +\end{corollary} + +Ještě budeme potřebovat následující větu, která ekvivalentně charakterizuje konvergenci v distribuci, její důkaz je však aktuálně nad naše schopnosti. + +\begin{theorem}[Levyho věta o spojitosti] + Platí $\vec X_n \overset D \to \vec X \Leftrightarrow \varphi_{\vec X_n}(\vec t) \to \varphi_{\vec X}(\vec t)$ pro všechna $\vec t \in \R^d$. +\end{theorem} + +Ukážeme si explicitní vyjádření charakteristické funkce normálního rozdělení. + +\begin{example} + Nechť $X \sim N(\mu, \sigma^2)$. Potom platí + $$\varphi_X(t) = \exp\left\{i\mu t - \frac{1}{2}\sigma^2 t^2\right\}.$$ +\end{example} + +\begin{proof} + Nejdříve uvažujme $X \sim N(0, 1)$ standardní normální rozdělení. Potom + $$ \varphi_X(t) = \E[\exp\{itx\}] = \E[\cos(tx)] + i\E[\sin(tx)] = $$ + $$ \int_{-\infty}^\infty \cos(tx)dP_X + i \int_{-\infty}^\infty \sin(tx)dP_X = \E[\cos(tX)], $$ + kde poslední rovnost plyne z faktu, že $\sin$ je lichá funkce, a tedy příslušný integrál je nulový. + Zderivujeme $\varphi_X(t)$ s použitím majoranty $x$. + $$ \odv*{\varphi_X(t)}{t} = - \int_{-\infty}^\infty x \sin(tx) dP_X = -\int_{-\infty}^\infty x \sin(tx) \frac{1}{\sqrt{2\pi}}\exp\left\{-\frac{x^2}{2}\right\} dx = $$ + $$ \overset{p.p.}= [x\sin(tx)f_X(x)]^\infty_{-\infty} - \int_{-\infty}^\infty t\cos(tx)f_X(x) dx = -t\E[\cos(tX)]. $$ + + Dostali jsme, že $\odv*{\varphi_X(t)}{t} = -t\varphi_X(t)$. Tato diferenciální rovnice s počáteční podmínkou $\varphi_X(0) = \cos 0 = 1$ má jediné řešení $\varphi_X(t) = \exp\left\{-\frac{t^2}{2}\right\}$. + + Potom pro $Y = \mu + \sigma X$ ($Y \sim N(\mu, \sigma^2)$) dosadíme a aplikujeme výsledek pro standardní rozdělení, dostaneme + $$ \varphi_Y(t) = \E[\exp\{itY\}] = E[\exp\{it(\mu + \sigma X\}) = $$ + $$ \exp\{it\mu\}\cdot\E[\exp\{i(t\sigma)X\} = \exp\left\{it\mu - \frac{1}{2}\sigma^2 t^2\right\}. $$ +\end{proof} + +\hfill \textit{konec 13. přednášky (31.3.2025)} diff --git a/stredni-hodnota.tex b/stredni-hodnota.tex index 16b917d..34b7bd7 100644 --- a/stredni-hodnota.tex +++ b/stredni-hodnota.tex @@ -273,12 +273,12 @@ Budeme pokračovat základními vlastnostmi variančních matic, které se chova \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 - $$ g(X) := e^\frac{-\varepsilon x}{2} + e^\frac{\varepsilon x}{2}, $$ + $$ g(x) := e^\frac{-\varepsilon x}{2} + e^\frac{\varepsilon x}{2}, $$ 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 $\int_\R x^mdP_X(x) =: \E X^m$. + Zbytek se dokáže indukcí s použitím podobné majoranty, v $m$-tém kroku dostaneme $\int_\R x^mdP_X(x) =: \E X^m$. Druhou vlastnost dokážeme přímým rozepsáním definice $$ \psi_Y(t) = \psi_{aX + b}(t) = \E[\exp(taX + tb)] = \E[\exp\{atX\}e^{tb}] = $$