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Je tedy $e^{ity} = 1 + ity + \frac{(ity)^2}{2} + o(t^2)$. Potom také + $\E[e^{itY}] = 1 + it\E Y - \frac{t^2}{2}\E Y^2 + o(t^2)$. + + Definujme $Y_n := \frac{(X_n - \E X_1)}{\sqrt{\Var X_1}}$. Pak $\E Y_n = 0, \Var Y_n = 1$ a platí + $$\frac{1}{\sqrt{n}} \sum_{i = 1}^n Y_i = \sqrt{n} \left(\frac{1}{n} \sum_{i=1}^n Y_i\right) = \sqrt{n}\bar Y_n = \sqrt{n}\frac{\bar X_n -\E X_1}{\sqrt{\Var X_1}} =: Z_n$$ + pro každé $n \in \N$. Dále můžeme počítat + $$ \varphi_{Z_n}(t) = \varphi_{\sum_{i=1}^n \frac{1}{\sqrt{n}} Y_i} (t) = \left(\varphi_{\frac{Y_1}{\sqrt{n}}}(t)\right)^n = \left(\varphi_{Y_1} \frac{t}{\sqrt{n}}\right)^n = $$ + $$ \left(1 + \frac{it}{\sqrt{n}}\E Y_1 - \frac{t^2}{2n} \E Y_1^2 + o\left(\frac{t^2}{n}\right)\right)^n = \left(1 - \frac{t^2}{2n} + o\left(\frac{t^2}{n}\right)\right)^n,$$ + neboť $\E Y_1 = 0$ a $\E Y_1^2 = \Var Y_1 = 1$. + Jelikož $e^y = \lim_{n \to \infty} \left( 1 + \frac{y}{n}\right)^n$, dostáváme + $$ \lim_{n \to \infty} \varphi_{Z_n}^t = \lim_{n\to\infty}\left(1 - \frac{t^2}{2n}\right)^n = e^{-\frac{1}{2}t^2} = \varphi_Z(t), $$ + kde $Z \sim N(0, 1)$. Z Lévyho věty (Věta \ref{thm-levy}) je konvergence charakteristických funkcí ekvivalentní konvergenci v distribuci, čímž je důkaz ukončen. +\end{proof} + +Následující věta nám umožní zformulovat mnohorozměrnou verzi centrální limitní věty. + +\begin{theorem}[Cramér-Woldova věta] + Platí $\vec X_n \overset D \to X$ právě tehdy, když pro všechna $\vec t \in \R^d$ platí $\vec t^T \vec X_n \overset D \to \vec t^T \vec X$. +\end{theorem} + +\begin{proof} + Důkaz plyne okamžitě z Věty \ref{thm-levy} a spojitosti lineárních zobrazení. +\end{proof} + +\begin{theorem}[Mnohorozměrná CLT] + Pokud $\{\vec X_n\}_{n \in \N}$ je IID posloupnost $d$-rozměrných náhodných vektorů s pozitivně definitní varianční-kovarianční maticí $\Var \vec X_1$, pak + $$ \sqrt{n}(\bar {\vec X }_n - \E \vec X_1) \overset D \to N_d(\vec 0, \Var \vec X_1), n \to \infty. $$ +\end{theorem} + +\begin{theorem}[Delta metoda] + Pokud $\sqrt{n}(Y_n - \mu) \overset D \to N(0, \sigma^2)$ a $g$ je spojitě diferencovatelná v okolí $\mu$ taková, že $g'(\mu) \neq 0$, pak + $$ \sqrt{n}(g(Y_n) - g(\mu)) \overset D \to N(0, (g'(\mu))^2\sigma^2). $$ +\end{theorem} + +\begin{proof} + Podle věty o střední hodnotě dostáváme $g(Y_n) = g(\mu) + g'(\tilde \mu)(Y_n - \mu)$, kde $\tilde \mu$ leží mezi $Y_n$ a $\mu$. Když $\sqrt{n}(Y_n - \mu) \overset D \to N(0, \sigma^2)$, potom $Y_n - \mu \overset D \to 0$. Pak tedy i $Y_n - \mu \overset P \to 0$. Jelikož $|\tilde \mu - \mu | \leq |Y_n - \mu|$, musí platit, že $\tilde \mu - \mu \overset P \to 0$. Dle věty o spojitém zobrazení (Věta \ref{thm-continuous-mapping}) dostáváme $g'(\tilde \mu) \overset P \to g(\mu)$ (předpokládáme spojitost $g'$). + + Dále můžeme dosadit $\sqrt{n}[g(Y_n) - g(\mu)] = g'(\tilde\mu)\sqrt{n}(Y_n - \mu)$. Nakonec, ze Slutského věty (Věta \ref{thm-slutsky}) máme (využili jsme předpoklad, že $\sqrt{n}(Y_n - \mu) \overset D \to N(0, \sigma^2)$) + $$ \sqrt{n}[g(Y_n) - g(\mu)] \overset D \to N(0, [g'(\mu)]^2 \sigma^2). $$ +\end{proof} + +Obdobně se dá zformulovat i mnohorozměrná delta-metoda (za předpokladu nenulovosti jakobiánu $g$). + +\hfill \textit{konec 14. přednášky (1.4.2025)} diff --git a/stochasticke-nerovnosti.tex b/stochasticke-nerovnosti.tex index 7f18045..839dc08 100644 --- a/stochasticke-nerovnosti.tex +++ b/stochasticke-nerovnosti.tex @@ -28,6 +28,7 @@ V této kapitole budeme studovat užitečné nerovnosti, které budeme moci apli \end{proof} \begin{theorem}[Čebyševova nerovnost] + \label{thm-chebyshev} Nechť $X$ je náhodná veličina a předpokládejme, že $\E[X]$ existuje, potom pro každé $\varepsilon > 0$ platí $$ P[|X - \E X| \geq \varepsilon] \leq \frac{\Var[X]}{\varepsilon^2}. $$ \end{theorem}