Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 201 / 3 / d / Solution

Split the adjacent products into two subsequences:Within each subsequence the pairs use disjoint coordinates, so each subsequence consists of independent and identically distributed random variables. The two subsequences need not be independent of each other. By independence within a pair,Apply the strong law of large numbers from (c) separately to obtain $k^{-1}{\sum }_{j=1}^k{Y}_j\to m^2$ and $k^{-1}{\sum }_{j=1}^k{Z}_j\to m^2$ simultaneously almost surely. Sinceand both prefactors tend to $1/2$, the strong law for adjacent products gives