Past exam of the probability course of the University of Oxford / 2023 / 1 / c / i / Solution

Direct integration of the uniform distribution yields $\mathbb{E}{U}_j=b/2$ and $\mathbb{E}{U}_j^2=b^2/3$. The expectation of a product of independent random variables givesSubtracting the square of the expected value gives the variance:Since $W_n\ge 0$, Markov's inequality implies, for every $\epsilon >0$,Hence $\boxed{W_n\to 0\text{ in probability for }0<b<2.}$ The first moment suffices: the variance need not tend to zero throughout this range.