 Fractional moment decay of independent mean-one products

Suppose $(X_n)$ are identically distributed independent random variables with $X_n\ge 0$, $\mathbb{E}{X}_n=1$, and $X_1$ not constant almost surely. For each $0<r<1$, strict concavity of $t^r$ givesThe moment is finite because $t^r\le 1+t$ on $[0,\infty )$. Let $Y_n={\prod }_{k=1}^n{X}_k$. Independence and Markov's inequality giveThe series of the latter probabilities is finite. The First Borel-Cantelli lemma, applied for the countable choices $\epsilon =1/m$, proves $Y_n\to 0$ almost surely. This proof requires neither existence nor integrability of $\log X_1$.

Although this nonnegative martingale has $\mathbb{E}{Y}_n=1$ at every time, its terminal expected value is zero. It is therefore not uniformly integrable. The contrast is useful: bounded expected values ensure almost sure convergence here, but do not allow preservation of the expected value through the limit.