I'm writing up a derivation of an expression for mutual information between weakly interacting Poisson processes. I'm running into an expression that looks like this:
$$\log\mathbb{E}\left[e^{\epsilon X + (\epsilon X)^2 + \log\left(1+\epsilon X + (\epsilon X)^2\right)}\right]$$
for a Poisson variable $X$ and a small parameter $\epsilon>0$. I want to expand this out to $O(\epsilon^2)$ and throw away higher-order terms, but I need to know that those terms are finite. If the $X^2$ term weren't there, I could expand the expression in terms of cumulants, but I'm not sure whether the moment-generating function for $X^2$ exists.
Is this expression finite, and if so, how can I demonstrate that?