fixed doob's lp inequality

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]