From 9019423e48f00ca1bd3a9e67f778489dc4747c0d Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Sat, 15 Jul 2023 02:00:04 +0200 Subject: [PATCH] d for convergence in distribution --- inputs/lecture_10.tex | 6 +++--- inputs/lecture_11.tex | 2 +- inputs/lecture_20.tex | 6 +++--- inputs/prerequisites.tex | 12 ++++++------ 4 files changed, 13 insertions(+), 13 deletions(-) diff --git a/inputs/lecture_10.tex b/inputs/lecture_10.tex index 83556c6..bc764fb 100644 --- a/inputs/lecture_10.tex +++ b/inputs/lecture_10.tex @@ -241,7 +241,7 @@ Unfortunately, we won't prove \autoref{thm:bochner} in this lecture. \begin{definition} We say that a series of random variables $X_n$ \vocab[Convergence!in distribution]{converges in distribution} - to $X$ (notation: $X_n \xrightarrow{\text{dist}} X$), iff + to $X$ (notation: $X_n \xrightarrow{\text{d}} X$), iff $\bP_n \implies \bP$, where $\bP_n$ is the distribution of $X_n$ and $\bP$ is the distribution of $X$. \end{definition} @@ -256,7 +256,7 @@ for all $f \in C_b(\R)$. \end{example} \begin{theorem} % Theorem 1 \label{lec10_thm1} - $X_n \xrightarrow{\text{dist}} X$ iff + $X_n \xrightarrow{\text{d}} X$ iff $F_n(t) \to F(t)$ for all continuity points $t$ of $F$. \end{theorem} % \begin{proof}\footnote{This proof was not done in the lecture, @@ -328,7 +328,7 @@ for all $f \in C_b(\R)$. % \end{proof} \begin{theorem}[Levy's continuity theorem]\label{levycontinuity} % Theorem 2 - $X_n \xrightarrow{\text{dist}} X$ iff + $X_n \xrightarrow{\text{d}} X$ iff $\phi_{X_n}(t) \to \phi(t)$ for all $t \in \R$. \end{theorem} We will assume these two theorems for now and derive the central limit theorem. diff --git a/inputs/lecture_11.tex b/inputs/lecture_11.tex index 5277228..06e49dd 100644 --- a/inputs/lecture_11.tex +++ b/inputs/lecture_11.tex @@ -59,7 +59,7 @@ and $\Var(\frac{S_n - \bE[S_n]}{\sqrt{\Var(S_n)}}) = 1$. Let $S_n \coloneqq \sum_{i=1}^n X_i$. Then \[ - \frac{S_n - n \nu}{\sigma \sqrt{n} } \xrightarrow{dist.} \cN(0,1), + \frac{S_n - n \nu}{\sigma \sqrt{n} } \xrightarrow{\text{d}} \cN(0,1), \] i.e.~$\forall x \in \R:$ \[ diff --git a/inputs/lecture_20.tex b/inputs/lecture_20.tex index 154db22..ade9eff 100644 --- a/inputs/lecture_20.tex +++ b/inputs/lecture_20.tex @@ -105,14 +105,14 @@ we need the following theorem, which we won't prove here: (Note that this argument does not work for $p = 1$, because $(L^\infty)^\ast \not\cong L^1$). - Let $A \in \cF_m$ for some fixed $m$ and write - $Y = \One_A$. + Let $A \in \cF_m$ for some fixed $m$ + and choose $Y = \One_A$. Then \begin{IEEEeqnarray*}{rCl} \int_A X \dif \bP &=& \lim_{k \to \infty} \int_A X_{n_k} \dif \bP\\ &=& \lim_{k \to \infty} \bE[X_{n_k} \One_A]\\ - &\overset{\text{for }n_k \ge m}{=}& \lim_{k \to \infty} \bE[X_m \One_A]. + &\overset{\text{for }n_k \ge m}{=}& \bE[X_m \One_A]. \end{IEEEeqnarray*} Hence $X_n = \bE[X | \cF_m]$ by the uniqueness of conditional expectation and by \autoref{ceismartingale}, diff --git a/inputs/prerequisites.tex b/inputs/prerequisites.tex index 9424694..9e6205c 100644 --- a/inputs/prerequisites.tex +++ b/inputs/prerequisites.tex @@ -37,7 +37,7 @@ from the lecture on stochastic. but has been added here for completeness; see \autoref{def:weakconvergence}. } - ($X_n \xrightarrow{\text{dist}} X$) + ($X_n \xrightarrow{\text{d}} X$) iff for every continuous, bounded $f: \R \to \R$ \[ \bE[f(X_n)] \xrightarrow{n \to \infty} \bE[f(X)]. @@ -58,7 +58,7 @@ from the lecture on stochastic. \begin{tikzpicture} \node at (0,1.5) (as) { $X_n \xrightarrow{\text{a.s.}} X$}; \node at (1.5,0) (p) { $X_n \xrightarrow{\bP} X$}; - \node at (1.5,-1.5) (w) { $X_n \xrightarrow{\text{dist}} X$}; + \node at (1.5,-1.5) (w) { $X_n \xrightarrow{\text{d}} X$}; %\node at (3,1.5) (L1) { $X_n \xrightarrow{L^1} X$}; \node at (3,1.5) (Lp) { $X_n \xrightarrow{L^p} X$}; \node at (3,3) (Lq) { $X_n \xrightarrow{L^q} X$}; @@ -121,7 +121,7 @@ from the lecture on stochastic. hence $X_n \xrightarrow{\bP} X$. \end{subproof} \begin{claim} %+ - $X_n \xrightarrow{\bP} X \implies X_n \xrightarrow{\text{dist}} X$. + $X_n \xrightarrow{\bP} X \implies X_n \xrightarrow{\text{d}} X$. \end{claim} \begin{subproof} Let $F$ be the distribution function of $X$ @@ -187,14 +187,14 @@ from the lecture on stochastic. \end{subproof} \begin{claim} - $X_n \xrightarrow{\text{dist}} X \notimplies X_n \xrightarrow{\bP} X$. + $X_n \xrightarrow{\text{d}} X \notimplies X_n \xrightarrow{\bP} X$. \end{claim} \begin{subproof} - Note that $X_n \xrightarrow{\text{dist}} X$ only makes a statement + Note that $X_n \xrightarrow{\text{d}} X$ only makes a statement about the distributions of $X$ and $X_1,X_2,\ldots$ For example, take some $p \in (0,1)$ and let $X$, $X_1, X_2,\ldots$ be i.i.d.~with $X \sim \Bin(1,p)$. - Trivially $X_n \xrightarrow{\text{dist}} X$. + Trivially $X_n \xrightarrow{\text{d}} X$. However \[ \bP[|X_n - X| = 1]