Wald's equation for a sum of functions of random variables

Question

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 ??

Answer 1

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.