= Almost-sure growth rate of independent products
{title2=$n^{-1}\log\prod_{j=1}^nU_j\to\mathbb E\log U_1$}
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 $b>e$.
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