Let $(X_k)_{k \in \mathbb{N}}$ be iid uniformly distributed on $(-1,1)$, $Y_0:=0$ and $$ Y_n := \sum_{k=0}^n \sin(\pi X_k). $$ We deine another random variable by $Z_K := \inf \lbrace n \in \mathbb{N}: \lvert Y_k \rvert \geq K \rbrace$. Here, $K>1$. I am interested in whether $Z_k$ is finite a.s..
I think that it is, since the series $\sum_{k=0}^n \sin(\pi X_k)$ diverges if $X_k \not \rightarrow 0$. But since the $X_k$ are iid uniformly distributed, this does not hold true. But this is not very rigorous. Ist this outline even correct and ist there an elegant proof for this?
Thank you!
Let $p$ be the probability that $X_1>0.5$. Note that $p$ is strictly positive.
Notice that, if there ever exist $\lceil 4K\rceil $ consecutive indices, $i$, for which $X_i>0.5$, then $Z_K$ will be finite. This is because it is impossible for the running total, $Y_n$, to stay within the interval $(-K,K)$, while moving a total distance of more than $\lceil 4K\rceil\cdot 0.5$ to the right over the course of these $\lceil 4K\rceil$ indices.
If we divide up the natural numbers into blocks of consecutive numbers with length $\lceil 4K\rceil$, then for each block, there is a probability of $p^{\lceil 4K\rceil}$ that $X_i>0.5$ for each $i$ in that block. Since the blocks are independent of each other, and $p^{\lceil 4K\rceil}$ is strictly positive, we are certain that there will be at least one block where this occurs. Therefore, $Z_K$ is almost surely finite.