Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 2 / 26J / d / Solution

{title2=$L^2$ convergence of averages of independent variables with bounded second moments}

For independent random variables with $\sup_n\mathbb E|X_n|^2<\infty$, write
$$
Y_N=\frac1N\sum_{k=1}^NX_k,
\qquad
m_N=\frac1N\sum_{k=1}^N\mathbb EX_k.
$$
Then $\lVert Y_N-m_N\rVert_2^2=N^{-2}\sum_{k\leq N}\operatorname{Var}(X_k)=O(N^{-1})$. Hence $Y_N$ converges in $L^2$ exactly when the scalar <Cesaro means> $m_N$ converge.

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