= Proof of Wald's equation by stopping indicators
{c}
If $M$ is a <stopping time> for independent identically distributed integrable variables $X_i$ and $\mathbb EM<\infty$, then $\{M\geq i\}$ depends only on $X_1,\ldots,X_{i-1}$ and is independent of $X_i$. Hence
$$
\mathbb E\sum_{i=1}^M X_i
=\sum_{i\geq1}\mathbb E[X_i\mathbf1_{\{M\geq i\}}]
=\mathbb EX_1\sum_{i\geq1}\mathbb P(M\geq i)
=\mathbb EX_1\,\mathbb EM.
$$
The interchange is justified because the same calculation with $|X_i|$ gives finite expected absolute sum.
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