I am trying to understand the following. If $M$ is the number of different random variables $X_1 ... X_M$, are there any conditions under which we can claim that:
$$E\left[f(\sum_{m=1}^M w_m X_m)\right] \leq \sum_{m=1}^M w_mE[f(X_m)]$$
So far, I think that it suffices for $f$ to be convex and for all $w_m$ to be non-negative with $\sum_{m=1}^M w_m = 1$. Because then, if we treat $E[f(.)]$ as itself a function, then this Jensen's formulation applies. However:
- I am not sure this is a reasonable interpretation to make when applying Jensen inequality to random variables
- if not, are there other conditions (for ex. bounds on the random variables or characteristics of $f$) that would make the inequality to hold?
Jensen inequality can be relevant here, but not wrt your expectation (because Jensen is about taking the expectation inside-outside the function, which is not the case here).
If we instead leave the expectation aside, then, if $f$ is convex and $w_m \ge 0$, letting $W=\sum w_w$, and we can indeed apply Jensen (again, ignoring that $X_m$ are random variables) :
$$f \left(\frac{1}{W} \sum w_m X_m\right) \le \frac{1}{W} \sum w_m f(X_m)$$
In particular, if $W=1$:
$$f \left(\ \sum w_m X_m\right) \le \sum w_m f(X_m)$$
Taking expectations on both sides you get your inequality.