From f097e9adb9e403d49c32707d1a0634296bcd71c0 Mon Sep 17 00:00:00 2001
From: Josia Pietsch <josia.pietsch@uni-bonn.de>
Date: Mon, 17 Jul 2023 20:59:06 +0200
Subject: [PATCH] s/sub-martingale/submartingale/g

---
 inputs/lecture_16.tex | 9 +++++----
 1 file changed, 5 insertions(+), 4 deletions(-)

diff --git a/inputs/lecture_16.tex b/inputs/lecture_16.tex
index dca129c..62dce27 100644
--- a/inputs/lecture_16.tex
+++ b/inputs/lecture_16.tex
@@ -183,21 +183,22 @@ Typically $\cF_n = \sigma(X_1, \ldots, X_n)$ for a sequence of random variables.
             for all $n$.
     \end{itemize}
 
-    $(X_n)_n$ is called a \vocab{sub-martingale},
+    $(X_n)_n$ is called a \vocab{submartingale},
     if it is adapted to $\cF_n$ but
     \[
     \bE[X_{n+1} | \cF_n] \overset{\text{a.s.}}{\ge} X_n.
     \]
-    It is called a \vocab{super-martingale}
+    It is called a \vocab{supermartingale}
     if it is adapted but $\bE[X_{n+1} | \cF_n] \overset{\text{a.s.}}{\le} X_n$.
 \end{definition}
 \begin{corollary}
+    \label{cor:convexmartingale}
     Suppose that $f: \R \to  \R$ is a convex function such that $f(X_n) \in L^1(\bP)$.
     Suppose that $(X_n)_n$ is a martingale%
     \footnote{In this form it means, that there is some filtration,
     that we don't explicitly specify}.
-    Then $(f(X_n))_n$ is a sub-martingale.
-    Likewise, if $f$ is concave, then $((f(X_n))_n$ is a super-martingale.
+    Then $(f(X_n))_n$ is a submartingale.
+    Likewise, if $f$ is concave, then $((f(X_n))_n$ is a supermartingale.
 \end{corollary}
 \begin{proof}
     Apply \autoref{cjensen}.