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.
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.
Best Answer
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.