Let $P$ be a probability measure on a space $\mathcal X$ and $h: \mathcal X \rightarrow \mathbb R$ is measurable function with finite moments of order 1 and 2. I'm interested in approximating the log MGF $t \mapsto \log \mathbb E_P[\exp(th(x)]$.

Now, using a simple Taylor expansion, one can carelessly write

$$ \log \mathbb E_P[\exp(th(x)] = t\mathbb E_P[h(x)] + \frac{t^2}{2} \operatorname{Var}_P[h(x)] + o(t^2). $$

# Question

Is the above representation formally correct ? If what are the formal bits missing ?

If $\mathcal X$ has metric structure and $h$ is $M$-Lipschitz, does anything change ? That is can more be said ?

- For example if $P$ is a 1-subgaussian distribution and $h$ is Lipschitz, can anything more be said about $\log \mathbb E_P[\exp(th(x)]$. For example, are interesting bounds on $\log \mathbb E_P[\exp(th(x)]$ is this case ? My rough guess is that $\log \mathbb E_P[\exp(th(x)] \le M^2t^2 / 2$, but I'm not sure.