Past exam of the year 1 of the computer science course of the University of Oxford / 2017 / Discrete mathematics, continuous mathematics and probability / 9 / c

Put $Z_i=Y_{i-1}{Y}_i$ for $1\le i\le n$. Independence of the underlying exponentials gives $\mathbb{E}{Z}_i=1$ andBy the covariance of overlapping independent products, adjacent terms have covariance one. Terms separated by at least two indices use disjoint independent factors, so their covariance is zero. There are $n-1$ adjacent pairs. Linearity of expectation and the variance of a sum formula therefore yieldIn particular, the products themselves are not independent merely because the original $Y_i$ are.

The normalized sum has mean one and variance $(5n-2)/n^2$. For every fixed $ϵ>0$, the Chebyshev inequality givesThis proves the requested limit, and in fact proves convergence in probability of $S_n/n$ to one. It is a finite covariance range weak law: adjacent dependence is harmless here because only linearly many covariance terms survive.