[Math] Product of i.i.d. random variables uniformly distributed on $(-1,1)$ converges almost surely to $0$

Question

Let $(\Omega,\mathcal{F},P)$ be a probability space and $(X_n)$ i.i.d. random variables with uniform distribution on $(-1,1)$. If $Y_n=X_1\ldots X_n$, prove that $(Y_n)$ converges to $0$ a.s.

Answer 1

By the Borel-Cantelli lemma, if the series $$ \sum_{n=1}^\infty P\{|Y_n|>\varepsilon\} $$ converges for each $\varepsilon>0$, $Y_n\to0$ almost surely as $n\to\infty$. Using Chebyshev's inequality, $$ \sum_{n=1}^\infty P\{|Y_n|>\varepsilon\} \le\frac1{\varepsilon^2}\sum_{n=1}^\infty\operatorname E|Y_n|^2 $$ since $\operatorname EY_n=0$. By independence, $$ \operatorname E|Y_n|^2=\operatorname E|X_1|^2\ldots\operatorname E|X_n|^2. $$ We have that $$ E|X_1|^2=\frac{(1-(-1))^2}{12}=\frac13. $$ Hence, $$ \sum_{n=1}^\infty P\{|Y_n|>\varepsilon\} \le\frac1{\varepsilon^2}\sum_{n=1}^\infty\biggl(\frac13\biggr)^n<\infty $$ for each $\varepsilon>0$ and $Y_n\to0$ almost surely as $n\to\infty$.