= Fractional moment criterion for convergence in probability
{title2=$\mathbb P(|Y_n|>\varepsilon)\leq\varepsilon^{-\alpha}\mathbb E|Y_n|^\alpha$}
For $\alpha>0$, <Markov's inequality> gives the displayed bound, so $\mathbb E|Y_n|^\alpha\to0$ implies <convergence in probability> to zero even when the first <absolute moment> does not tend to zero. For products $Y_n=\prod_{j=1}^nU_j$ of <independent and identically distributed> factors, it suffices that $\mathbb E|U_1|^\alpha<1$. With the <uniform distribution> on $(0,b)$, the <product of independent uniform random variables> formula makes this $\alpha\log b<\log(1+\alpha)$. Since $\log(1+\alpha)/\alpha\to1$ as $\alpha\downarrow0$, such an $\alpha$ exists whenever $b<e$.
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