Convergence of sum of exponential of random walk

Question

I am trying to solve the question:
Let $X_1,X_2,\dots$ be i.i.d., $\mathbb{P}(X_1=1)=\mathbb{P}(X_1=-1)=1/2$, and $S_n=X_1+\dots+X_n$.
Prove that the following random variable converges in distribution as $n\to\infty$, and identify the limit:
$$
\left(\sum_{k=1}^ne^{S_k}\right)^{\frac{1}{\sqrt{n}}}
$$
Thoughts: It is sufficient (right?) to show the convergence of $\frac{1}{\sqrt{n}}\log \sum_{k=1}^ne^{S_k}$. My guess is that it goes to $0$ (so that the required limit is $e^0=1$). I found a paper discussing the convergence of $\frac{1}{\sqrt{n}}\sum_{k=1}^nf({S_k})$ but there isn't a complete version online. Perhaps law of iterated logarithm might help?

Answer 1

Let $M_n$ be the maximum of $S_1,\dots,S_n$ and let $Z$ be a standard normal random variable. It is a well known consequence of the reflection principle and the CLT that $M_n/\sqrt{n}$ tends to $|Z|$ in law; it also follows from Donsker's invariance principle, see Theorem 5.25 page 134 in [1]. Observe that $$ e^{M_n} \le \sum_{k=1}^n e^{S_k} \le n e^{M_n} $$ so $$ {M_n} \le \log\left(\sum_{k=1}^n e^{S_k}\right) \le {M_n}+ \log n \,. $$ Thus $$ {\frac{1}{\sqrt{n}}} \log\left(\sum_{k=1}^n e^{S_k}\right) $$ tends in distribution to $|Z|$, whence $$ \left(\sum_{k=1}^ne^{S_k}\right)^{\frac{1}{\sqrt{n}}} $$ tends in distribution to $e^{|Z|}$.

[1] Mörters, Peter, and Yuval Peres. Brownian motion. Vol. 30. Cambridge University Press, 2010. https://www.yuval-peres-books.com/brownian-motion/