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Is there theorem similar to Lévy's continuity theorem, but considering moment generating functions (mgf) instead of characteristic functions?

For example, suppose $M_n(t) = \mathbb E e^{tX_n}$ are mgf of $X_n$, which are all finite on $(-\varepsilon, \varepsilon)$ for some fixed $\varepsilon > 0$. Suppose the $\lim_{n\to\infty}M_n(t) = M(t)$ exists for all $t \in (-\varepsilon, \varepsilon)$, and $M(t)$ is continuous. Could we conclude that $M$ is mgf, and $X_n$ converge in distribution to $X$ with distribution characterized by $M$?

Moment generating functions and characteristic functions are quite similar in definition. On top of that, once they both exist, each of them determines distribution. It seems reasonable to have analogue theorem for mgf.

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Yes. Here is a statement and proof for Laplace transforms in general (Theorem 2.1 (ii) and in particular for your case Corollary 2.2). In the notes, your $t$ is his $-\lambda$. Note also that the probability measures $\mu_n$ corresponding to your random variables are defined by $\mu_n (A) = P(X_n \in A)$ for all $A$ in the relevant sigma algebra. I hope this answers your question.

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