Let $X_1, X_2, \dots$ be iid, non-negative random variables with mean 1, and define the martingale $M_n$ to be the product of the first $n$ terms.
I want to show that if $M_n$ converges in $L^1$, then $X_1 = 1$ a.s. I'm thinking it might have something to do with uniform integrability, but it's not clear to me how to go about it.
Best Answer
Convergence of $M_n$ in $L^1$ implies $E[|M_{n+1}-M_n|]\to 0$, which implies $E[|X_{n+1}-1|]\to 0.$ That is the constant series $E[|X_{1}-1|]= 0$ must equal 0, which is the assertion.