Let $θ\sim\text{Unif}([0,2\pi])$ and define $X_k=\sin(kθ)$, $k\in\Bbb N$. Show that $(X_1+⋯+X_n)/n→0$ a.e Let $θ$ be uniformly distributed on $[0, 2\pi]$ and define $X_k= \sin k\theta 
(k= 1,2, ...)$ . Show that $(X_1+\cdots+X_n)/n\to0$ a.e
Attempts:Let $S_n=(X_1+\cdots+X_n)/n$,from Kolmogorov's Inequality we get for any $\epsilon>0,P\{\max\limits_{1\leq j\leq n}S_j\geq\epsilon\}\leq 1/(n\epsilon^2)$,but how to get $P\{\lim_n S_n\neq0\}=0$?
 A: This follows from Weyl's equidistribution theorem.
Since $\theta$ is uniformly distributed in $[0,2\pi]$,  $\mathbb{P}[\theta/2\pi\in\mathbb{Q}^c]=1$. Since $\sin$ is Riemann integrable in in $[0,2\pi]$, Weyl's theorem implies that
the average $\frac{X_1(\theta)+\ldots + X_n(\theta)}{n}$ converges to $\frac{1}{2\pi}\int^{2\pi}_0\sin(t)\,dt=0$.


*

*The fact that $\theta k/2\pi$ is equidistributed on $[0,1]$ $\mod 1$ when $\theta/2\pi$ is irrational is a well known fact and has been discussed in MSE before. Many books of Probability discuss this example in their treatment of ergodic theory (see Billingsley, P.,  Probability and Measure, third edition, New York, Wiley  1995, or Durrett, Probability: Theory and Examples, fifth edition, Cambridge University Press, 2019.)


*Weyl's equidistribution theorem can be studied without use of heavy machinery from Ergodic theory by means of Calculus (Riemann integration, and Weierstrass density theorem)

Note that that the random variables $X_n(\theta)=\sin(k\theta)$ have mean $0$ and are orthogonal:
$$\frac1{2\pi}\int^{2\pi}_0\sin(n\theta)\,d\theta=0,\qquad \frac1{2\pi}\int^{2\pi}_0\sin(n\theta)\sin(m\theta)\,d\theta=\frac12\delta_{n,m}$$
Hence a weak law of large numbers holds, that is, let $S_n=X_1+\ldots+X_n$. Then
$$\mathbb{E}\Big[\Big|\frac{S_n}{n}\Big|^2\Big]=\frac{1}{2n}\xrightarrow{n\rightarrow\infty}0$$
Thus $\frac1nS_n$ converges to $0$ in $L_2$, in probability and thus, almost surely through a subsequence.
