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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 ?

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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.

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  • $\begingroup$ **whence (use Chebyshev's Inequality, for instance) most of the probability of $Y_n$eventually is focused near zero.** how are you, sir, implementing this step ?? how are you using Chebyshev's inequality to show $Y_n$ converges to $0$ ? $\endgroup$ Jun 25, 2016 at 4:32

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