Let $\{X_{n}\}_{n \geq 1}$ be a sequence of iid random variables with $\mathbb E[|X_1|^\alpha] < \infty$ for some $\alpha > 0$. Derive a necessary and sufficient condition on $\alpha$ for almost sure convergence of the series $\sum_{n=1}^{\infty}X_{n}\sin(2\pi nt)$ for all $t \in (0, 1)$.
I tried to prove that for any $\epsilon > 0$, the sequence of partial sums forms a Cauchy sequence in $L^\alpha$ space. But that didn't help.