This is Exercise 5.2.9 from Durrett's Probability, Theory and Examples (4th ed.):
Let $Y_1, Y_2, \dotsc$ be nonnegative i.i.d. random variables with $E(Y_m) = 1$ and $P(Y_m = 1) < 1$. Show that:
- $X_n = \prod_{m \leq n} Y_m$ defines a martingale
- $X_n \to 0$ almost surely
- $(1/n) \log X_n \to c < 0$ (using the strong law of large numbers)
I believe I have answered parts (1) and (2). The first part is a straightforward verification. In the second part I use the martingale convergence theorem to state that the limit $X = \lim_{n \to \infty} X_n$ exists almost surely and has finite expectation; I then argue that there are positive numbers $a<b$ such that $$ P \left( a < X_n < b \text{ eventually} \right) = 0 , $$ where "$E_n$ eventually'' means $\bigcup_{n \geq 1} \bigcap_{m \geq n} E_m$. From here I conclude that $X_n \to 0$ almost surely, after a couple more steps.
However, I am having trouble with part (3). If we can show that $E(\log Y_1) = c < 0$, then the result follows from the strong law of large numbers. However, I am not quite sure how to establish this result. One thing I noticed is that by Jensen's inequality (since $-\log x$ is a convex function), $$ E(-\log Y_1) \geq -\log E(Y_1) = 0 , $$ which implies that $E(\log Y_1) \leq 0$. However, I could not figure out how to make this inequality strict. Any hints/suggestions?