If $X_1, X_2,...,X_n$ are independent and have the same expected value, but differing variances, does Wald's equation (that is, $E[X_i]=E[N] E[X]$) still apply?
In particular, my problem to find the distribution of $$\sum_{i=1}^n (-1)^iX_i $$ given the moment generating function for $X_m$.
I don't think that Wald's equation does apply here. I have calculated $X_m$ to be normally distributed with mean $0$ and variance $2m$. Where to go from here?
Yes, as Ben says in the comments, the question is why would Wald's equation apply here? It is for situations where the number of summands is random, and there's no indication that's the case here.
You can write the MGF of the alternating sum in terms of the MGFs for the $X_i$ as follows: $$ E(e^{t\sum_{i=1}^n(-1)^i X_i}) = \prod_{i=1}^nE(e^{t(-1)^iX_i})= \prod_{i=1}^nM_{X_i}((-t)^i)$$ where you use independence to factor the expected value.