= Integrability of a stopped random-walk increment
{title2=$\mathbb E|X_T|\leq\mathbb E|X_1|\,\mathbb ET$}
Let $X_n$ be integrable <independent and identically distributed random variables>, and let $T\geq1$ be a <stopping time> for their <natural filtration>, with $\mathbb ET<\infty$. Since $\{T\geq n\}$ is measurable before $X_n$ is observed, <independence> and the <Tonelli theorem> give $\mathbb E|X_T|\leq\sum_n\mathbb E[|X_n|\mathbf1_{\{T\geq n\}}]=\mathbb E|X_1|\mathbb ET$. The event $T=n$ need not be independent of $X_n$, so the selected increment can be biased.
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