From 9884449627612a74c7957441698360f90d812f90 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Mon, 17 Jul 2023 20:56:30 +0200 Subject: [PATCH] fixed doob's lp inequality --- inputs/lecture_18.tex | 11 +++++++---- 1 file changed, 7 insertions(+), 4 deletions(-) diff --git a/inputs/lecture_18.tex b/inputs/lecture_18.tex index dc9e110..11fafab 100644 --- a/inputs/lecture_18.tex +++ b/inputs/lecture_18.tex @@ -143,7 +143,8 @@ Now let $p \ge 1$ be not necessarily $2$. First, we need a very important inequality: \begin{theorem}[Doob's $L^p$ inequality] \label{dooblp} - Suppose that $(X_n)_n$ is a sub-martingale. + Suppose that $(X_n)_n$ is a martingale + or a non-negative submartingale. Let $X_n^\ast \coloneqq \max \{|X_1|, |X_2|, \ldots, |X_n|\}$ denote the \vocab{running maximum}. \begin{enumerate}[(1)] @@ -159,7 +160,7 @@ First, we need a very important inequality: In order to prove \autoref{dooblp}, we first need \begin{lemma} \label{dooplplemma} - Let $p > 1$ and $X,Y$ non-negative random variable + Let $p > 1$ and $X,Y$ non-negative random variables such that \[ \forall \ell > 0 .~ \bP[Y \ge \ell] \le @@ -213,8 +214,10 @@ In order to prove \autoref{dooblp}, we first need \bP[E_j] \overset{\text{Markov}}{\le } \frac{1}{\ell} \int_{E_j} |X_j| \dif \bP \label{lec18eq2star} \end{equation} - Since $(X_n)_n$ is a sub-martingale, $(|X_n|)_n$ is also a sub-martingale - (by \autoref{cjensen}). + We have that $(|X_n|)_n$ is a submartingale, + by \autoref{cor:convexmartingale} + in the case of $X_n$ being a martingale + and trivially if $X_n$ is non-negative. Hence \begin{IEEEeqnarray*}{rCl} \bE[\One_{E_j}(|X_n| - |X_{j}|) | \cF_j]