Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 26K / iii / Solution

The independent standard Gaussian random variables have mean zero and variance one. For $m>n$, the mixed products have expectation zero, givinguniformly over $m>n$ as $n\to \infty$. Thus $(Y_n)$ is Cauchy in the L2 space. Its completeness, proved in (i), suppliesMoreover $L^2$ convergence implies convergence of the means and second moments: $\mathbb{E}Y=0$ and $\mathbb{E}{Y}^2={\lim }_n{\sum }_{j\le n}{\alpha }_j^2={\sigma }^2$.