Let $X_1, X_2,\ldots $ be independent and identically distributed, positive random variables with $E(X_1) = 1$. Set $Y_n = \prod_{j=1}^n X_j$. I already showed that $Y_n$ is a martingale and converges almost surely to some random variable $Y$.

Moreover, the distribution of $X_j$ is given by $$ P(X_j = \frac12) = P(X_j = \frac32) = \frac12. $$ It requires to show that $Y_n \to 0$ almost surely.

I managed to show $$ \frac1n \sum_{j=1}^n \log(X_j) \to \frac12 \log3-\log2 \quad a.s. $$ but don't know what to do next...

Any thoughts?

  • 1
    $\begingroup$ If $s_n/n\to\ell$ with $\ell<0$ then $s_n\to-\infty$ (nothing stochastic). Can you prove this? $\endgroup$ – Did Oct 4 '15 at 14:15

Note that $$\frac{1}{2} \log 3 - \log 2 < 0$$ and therefore $$\log \left( \prod_{j=1}^n X_j \right) = \sum_{j=1}^n \log (X_j) = n \bigg(\underbrace{\frac{1}{n} \sum_{j=1}^n \log(X_j)}_{\to \frac{1}{2} \log 3 - \log 2 <0} \bigg) \to - \infty$$ almost surely. Now the continuity of $\exp$ entails

$$\prod_{j=1}^n X_j = \exp \left[ \log \left( \prod_{j=1}^n X_j \right) \right] \to 0 \qquad \text{a.s.}$$


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