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.

  • $\begingroup$ Did you try Kolomogorov's Three series Theorem? $\endgroup$ Commented Jul 30, 2023 at 7:15
  • $\begingroup$ I tried taking c = 1 and random variable to be X_{n}sin2*pint. Couldn't get anywhere. @geetha290krm. All I could conclude was it has to be the case that E|X_{1}| < \infty. But nothing further. $\endgroup$
    – Debu
    Commented Jul 30, 2023 at 7:21
  • $\begingroup$ any hint would be appreciated. @geetha290krm $\endgroup$
    – Debu
    Commented Jul 30, 2023 at 7:44


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