Suppose $\alpha>0$, $\xi_n$ are independent standard gaussians, $S_n = \sum_{k=1}^{n} \sin(\frac{\xi_k}{k^\alpha})$,
(a) Find all $\alpha$ such that $S_n$ converges with probability 1.
(b) If $\alpha$ is chosen such that $S_n$ is convergent with probability 0, find $a_n$ and $b_n$ such that $\frac{S_n-a_n}{b_n}$ converges to a non degenerate distribution $Z$.
For part (a), I have managed to show that the series converges with probability one if $\alpha>1$. But I have no idea if this condition is necessary. For the second part I guess Lindeberg condition applies? Any help is greatly appreciated!
For the first part, you can use the three series theorem. The convergence will hold if and only if $\sum_{k=1}^\infty \mathbb E\left[\sin^2\left(\xi_1k^{-\alpha}\right)\right]$ is finite. By the dominated convergence theorem combined $\sin(x)\leqslant x$ gives that $\left(S_n\right)$ converges with probability one if and only if $\alpha>1/2$.
Indeed, for the second part, Lindeberg's condition works. You have to find an equivalent for $\operatorname{Var}(S_n)$ to conclude.