I am looking for good solutions to this problem. Could you please help me with this? Any solution would be appreciated.
Let $\mathit{X_n}$ be independent and identically distributed. Assume $\mathit{X_1}$ is symmetric in the sense that $\mathit{X_1}$ and $\mathit{-X_1}$ have the same distribution. Let $\mathit{Y_n} = \mathit{X_n}1_{[|\mathit{X_n}|\le1]}$. Show $\frac{\sum_{i=1}^n \mathit{Y_i}}{n} \to 0$ almost surely.
We note $Y$ the r.v. such that $Y_n \sim Y$ for all $n$, same for $X$ relatively to $X_n$. $Y$ is bounded a.s., hence $E(|Y|)$ is finite. Since $X$ is symmetric, $Y$ is also symmetric and thus $E(Y)=0$.
By the strong law of large numbers : $$\frac{\sum_{i=1}^n \mathit{Y_i}}{n} \to E(Y) = 0 \text{ a.s. }$$