Martingale related question

Question

Let $S_n = Z_1 + · · · + Z_n$ where ${Z_k}$ are i.i.d. $N (0, 1)$ variables. Find constants $c_n$ such that $M_n = e^{S_n+c_n}$ is a martingale. How can I show using martingale theory that there exists a finite random variable $M_{\infty}$ such that $M_n → M_{\infty}$ a.s.

Answer 1

$M_n$ is nonnegative. Hence $\lim M_n$ exists a.s.

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From Probability with Martingales:

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