Fractional moment criterion for convergence in probability ($\mathbb{P}(|Y_n|>\epsilon )\le {\epsilon }^{-\alpha }\mathbb{E}|Y_n{|}^{\alpha }$)

For $\alpha >0$, Markov's inequality gives the displayed bound, so $\mathbb{E}|Y_n{|}^{\alpha }\to 0$ implies convergence in probability to zero even when the first absolute moment does not tend to zero. For products $Y_n={\prod }_{j=1}^n{U}_j$ 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 $\log (1+\alpha )/\alpha \to 1$ as $\alpha \downarrow 0$, such an $\alpha$ exists whenever $b<e$.