Almost sure convergence of product of i.i.d variables.

Question

Here is a problem I have not been able to solve for quite some time.

Let $\{Y_k\}_{k}$ be a sequence of non-negative, i.i.d random variables with $EY_k=1$ and $P(Y_k=1)<1.$ Then, $X_n = Y_1Y_2...Y_n$ is a martingale and converges a.s to $0.$

It can be easily shown that the $X_n$ is a martingale, but establishing a.s convergence proved to be tricky. The only sufficient condition for a.s convergence that I know of is the following:

$$\sum_{n=1}^{\infty}P(X_n>\varepsilon)<\infty$$ for arbitrary $\varepsilon>0.$ But I have been unable to found a sharp bound for $P(X_n>\varepsilon)$, so that the above series is convergent.

Any hint is appreciated.

Answer 1

HINT

Consider $\log(X_n).$ It is a sum of i.i.d. random variables with expectation less than zero.