I pulled this question from an old set of lecture notes. It goes as follows:
Suppose $Y_n$ are i.i.d random variables on a probability space $(\Omega, \mathcal{F}, P)$ such that $P(Y_n = \pm 1) = \frac{1}{2}$ and $X_n = \sum_{j=1}^n a_j Y_j$ where $\{a_n\} \subset \mathbb{R}_+$ is non-increasing (i.e. a positive real-valued non-increasing sequence). Show the following:
a) If $\sum_{j=1}^\infty a_j < \infty$, then there exists $X \in L^2$ such that $X_n \to X$ almost surely and in $L^2$.
b) Now suppose there exists a real-valued $X$. Show that $\sum_{j=1}^\infty a_j < \infty$ and, hence there exists an $X \in L^2$ such that $X_n \to X$ (Hint: consider $e^{X_n}$).
I understand how to do part a), mainly by showing that $X_n$ is a martingale with respect to the natural filtration $\mathcal{F}_n = \sigma(Y_1, Y_2, \ldots, Y_n)$. Next, by showing $\text{sup}_{n \in N} E[X_n^2] < \infty$, we get the result from one of the martingale convergence theorems.
My problem is I don't understand what b) is asking for. The question wording to me is strange, and I'm not sure what it is asking me to solve. These old notes are prone to some big typos, so I thought I would see if anyone perhaps could see what was being asked here. If you have any thoughts/corrections, I would appreciate hearing them.