I was reading the the proof of CLT by MGF in this answer
Let $Y_i$'s be i.i.d random variables with mean 0 and variance 1, and in the original answer, by Taylor expansion we have
$$M_{Y_1}(s) = E[\exp(sY_1)] = 1 + s E[Y_1] + \frac{s^2}{2} E[Y_1^2] + s^2 h(s) = 1 + \frac{s^2}{2} + s^2 h(s), \qquad \text{where $h(s) \to 0$ as $s \to 0$},$$
I wonder if there is a minimal or at least pretty general assumption we can impose on $Y_1$ to guarantee $h(s)$ indeed goes to zero as $s\to 0$. One sufficient assumption is that all moments of $Y_1$ are bounded by a same number. But this condition is too strong, as standard normal distribution itself does not have a uniform bound on its moments. Can anyone help me finding a better assumption on $Y_1$? Is it just enough to assume the existence of MGF on a domain to guarantee $h(s)\to 0$?
I know the general proof uses characteristic functions. But I only want to consider the proof by MGF.
