If $X_1,X_2,\dots$ are independent, $S_n=X_1+\cdots+X_n$ and $S_n\to S_\infty$ a.s., $\varphi_{S_\infty}(t)=\prod_{j=1}^\infty\varphi_{X_j}(t)$

Question

Problem: Let $X_1,X_2,\dots$ be independent and let $S_n=X_1+\cdots+X_n$. Suppose that $S_n\to S_\infty$ almost surely. Then $S_\infty$ has characteristic function $\prod_{j=1}^\infty\varphi_{X_j}(t).$

My Thoughts: Since the random variables are independent, we have that $$\varphi_{S_n}(t)=\prod_{j=1}^n\varphi_{X_j}(t).$$ That $S_n\to S_\infty$ almost surely implies that $S_n\to S_\infty$ in distribution as well, so that $\varphi_{S_n}(t)\to\varphi_{S_\infty}(t)$ for all $t\in\mathbb R$. But $$\varphi_{S_\infty}(t)=\lim_{n\to\infty}\varphi_{S_n}(t)=\lim_{n\to\infty}\prod_{j=1}^n\varphi_{X_j}(t)=\prod_{j=1}^\infty\varphi_{X_j}(t),$$ so we are done.

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I think this solution I have cooked up is too easy to be a problem in Rick Durrett's Probability: Theory and Examples. Therefore, I would like to ask if anyone can spot any mistakes in the above argument.

Thank you for your time and feedback.

Answer 1

As pointed out by Kavi Rama Murthy, your thoughts are correct; not all the problems in Durrett's book are hard, although a large part of them are tricky.