Almost-sure growth rate of independent products ($n^{-1}\log {\prod }_{j=1}^n{U}_j\to \mathbb{E}\log U_1$)

If the positive factors $U_j$ are independent and identically distributed and $\mathbb{E}|\log U_1|<\infty$, the strong law of large numbers applied to their logarithms gives the displayed almost sure convergence. A negative limit forces the products to tend to zero; a positive limit forces them to tend to infinity. A zero limit alone gives neither conclusion. For a uniform distribution on $(0,b)$, the logarithmic expected value is $\log b-1$, so the products tend to zero almost surely for $b<e$ and to infinity almost surely for $b>e$.