Continuous uniform random variables convergence question

Question

Let $X_1, X_2, \ldots$ be independent $U(0,2)$ random variables and let $$Y_n = \prod_{i=1}^n \, X_i \;.$$

How do I prove or disprove that that $Y_n$ converges to $0$ almost surely ?

Answer 1

Use the fact that $-\log(X_i/2)$ has an Exponential distribution. (This is easily proven.) The reasoning will proceed as follows. Justify each step:

• $-\log(Y_n) + n \log(2)$ is the sum of $n$ independent Exponential distributions,

• which therefore has a Gamma$(n)$ distribution,

• whose mean and variance are both $n$,

• whence (use Chebyshev's Inequality, for instance) most of the probability of $Y_n$ eventually is focused near zero.

Incidentally, it is insightful to contemplate the role played by the upper limit of $2$ in this question. When you have worked through the details, you will be able to see how the answer changes when $2=\exp(\exp(-0.3665129))$ is replaced by any value of $e = \exp(\exp(0))$ or greater.