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