TITLE: Convergence of sums of $\sin(kY)$ where Y is uniform
QUESTION [5 upvotes]: Let $Y$ be uniformly distributed on $[0,2\pi]$. Let $X_k=\sin(kY)$.
I want to show that $\frac{X_1+X_2+\cdots+X_n}{n}\to 0$ almost surely.
I thought of using the strong law of large numbers but it requires the $X_k$'s to be identically distributed.

REPLY [8 votes]: You can compute the geometric sum  $e^{iy}+e^{2iy}+..._+e^{iny}$ and show that $\frac 1 n (e^{iy}+e^{2iy}+..._+e^{iny}) \to 0$ for all real $y$ such that $e^{iy} \neq 1$. Take the imaginary part.  Can you finish?