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Suppose that we have an i.i.d. sequence of random varaible $(X_n)_{n\in \mathbb{N}}$, $M$ is a stopping time for $S_n = \sum_{i=1}^n X_i$, and that all the assumption for the validity of Wald's equations are verified so that

$\mathbb{E}\left\{\sum_{i=1}^M X_n\right\} = \mathbb{E}\left\{M\right\}\mathbb{E}\left\{X\right\}$.

If $f$ is a continuous, deterministic function so that $f(X)$ is always limited, the following

$\mathbb{E}\left\{\sum_{i=1}^M f(X_n)\right\} = \mathbb{E}\left\{M\right\}\mathbb{E}\left\{f(X)\right\}$

is valid ??

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If the random variable $f(X_1)$ is integrable (beware it is not enough to be finite, as your use of the word "limited" may suggest), the sequences $(f(X_n))_n$ will still form an i.i.d. sequence of random variables, and $T$ will again be a stopping time for the same filtration you started with, so yes, the theorem will apply likewise.

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  • $\begingroup$ Thank you for your answer, the function is actually integrable. $\endgroup$ – m696p Jul 9 at 11:39

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