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\infty} \P(|Y_n - y| > \delta) &= 0\\ + \lim\limits_{n \to \infty} \lim\limits_{m \to \infty} \P(|Z_n - z| > \delta) &= 0 + \end{align*} + Da $h$ eine stetige Funktion ist und $y$ und $z$ bereits feststehen gilt + \begin{align*} + \forall \epsilon > 0 \exists \delta > 0: \lVert(Y_n, Z_n) - (y,z)\rVert_1 \leq \delta &\implies |h(Y_n, Z_n) - h(y, z)| \leq \epsilon\\ + |Y_n - y| + |Z_n -z| \leq \delta &\implies |h(Y_n, Z_n) - h(y, z)| \leq \epsilon\\ + \{|h(Y_n, Z_n) - h(y, z)| \leq \epsilon\}&\supset \{|Y_n - y| + |Z_n -z| \leq \delta\}\\ + \{|h(Y_n, Z_n) - h(y, z)| > \epsilon\}&\subset \{|Y_n - y| + |Z_n -z| > \delta\}\\ + \P(|h(Y_n, Z_n) - h(y, z)| > \epsilon) &\leq P(|Y_n - y| + |Z_n -z| > \delta)\\ + \P(|h(Y_n, Z_n) - h(y, z)| > \epsilon) &\leq P(|Y_n - y| > \delta) + \P(|Z_n -z| > \delta) + \intertext{$Y_n \xrightarrow{\P} y$,$Z_n \xrightarrow{\P} z$} + \lim\limits_{n \to \infty} \P(|h(Y_n, Z_n) - h(y, z)| > \epsilon) &= 0. + \end{align*} + \item Auch $(a_n)_{n\in \N}$ kann als eine Folge von (konstanten) Zufallsvariablen aufgefasst werden. + Weil $h(a,X) = aX$ eine stetige Funktion ist, gilt $a_nX_n \xrightarrow{\P} aX$. + Weil $h(X, Y) = X + Y$ eine stetige Funktion ist, gilt $a_nX_n + Y_n \xrightarrow{\P} aX +Y$. + \end{enumerate} +\end{aufgabe} \begin{aufgabe} Sei $X, X_n\colon \Omega \to \R$ für $n \in \N$. @@ -105,4 +129,46 @@ \end{enumerate} \end{aufgabe} +\begin{aufgabe} + \begin{enumerate}[(a)] + \item Es gilt für $ 0 <\epsilon < 1$ + \begin{align*} + \lim\limits_{n \to \infty} \P(X_n > \epsilon) &= \lim\limits_{n \to \infty} \P(\sqrt{n}\mathbbm{1}_{[0,\frac{1}{n}]}(U) > \epsilon)\\ + &= \lim\limits_{n \to \infty} \P(\mathbbm{1}_{[0,\frac{1}{n}]}(U))\\ + &= \lim\limits_{n \to \infty} \frac{1}{n}\\ + &= 0. + \end{align*} + Also gilt $X_n \xrightarrow{\P} 0$ + Gleichzeitig erhalten wir + \begin{align*} + \E(|X_n|^2) &= \lim\limits_{n \to \infty} \int_\R (X_n)^2 \P(\d{x}) \\ + &= \lim\limits_{n \to \infty} \int_\R n \mathbbm{1}_{[0,\frac{1}{n}]}(U)\P(\d{x})\\ + &= \lim\limits_{n \to \infty} n \frac{1}{n}\\ + &= 1\\ + &\neq 0. + \end{align*} + Daraus folgt $X_n \not \xrightarrow{L^2} 0$. + \item Es gilt + \begin{align*} + \E(|X - X_n|^2) &= \E(|X- X_n|^2 \mathbbm{1}_{|X_n-X| > \epsilon}) + \E(|X- X_n|^2 \mathbbm{1}_{|X_n-X| \leq \epsilon})\\ + \intertext{Wir nutzen die Hölder-Ungleichung $\E(|X_nX|) \leq \sqrt{\E(|X|^2)\E(|X_n|^2)}$ und erhalten} + &= \E(|X|^2\mathbbm{1}_{|X_n-X| > \epsilon}) + \E(|X_n|^2\mathbbm{1}_{|X_n-X| > \epsilon}) - 2\E(|XX_n|\mathbbm{1}_{|X_n-X| > \epsilon}) + \E(|X- X_n|^2 \mathbbm{1}_{|X_n-X| \leq \epsilon}) + \intertext{Wegen $X_n \xrightarrow{\P} X$ ist $\{|X_n - X| > \epsilon\}$ eine Nullmenge und es gilt} + &= 0 + \E(|X- X_n|^2 \mathbbm{1}_{|X_n-X| \leq \epsilon})\\ + &= \epsilon^2 \E(\mathbbm{1}_{|X_n-X| \leq \epsilon})\\ + &= \epsilon^2 (1 - \P(|X_n - X| > \epsilon))\\ + &= \epsilon^2 + \end{align*} + Für $\epsilon \to 0$ erhalten wir daraus die Behauptung. + \item Betrachte + \begin{align*} + \limsup\limits_{n \to \infty} \E(|X_n|^{2 + \alpha}) &= \limsup\limits_{n \to \infty} \int_\R \sqrt{n}^{2 + \alpha} \cdot \mathbbm{1}_{[0,1]}(U) \P^U(\d x)\\ + &= \limsup\limits_{n \to \infty} n \cdot n^{\frac{\alpha}{2}} \cdot \frac{1}{n}\\ + &= \limsup\limits_{n \to \infty} n^{\frac{\alpha}{2}}\\ + &= \infty + \end{align*} + + \end{enumerate} +\end{aufgabe} + \end{document}