Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 26K / a

Assume $X_n\to X$ in $L^2$. The reverse triangle inequality for the $L^2$ norm givesSquaring the norms yields

Now assume $X_n\to X$ almost surely and $\mathbb{E}[X_n^2]\to \mathbb{E}[X^2]$. In particular, $(X_n)$ is bounded in $L^2$, so it is uniformly integrable in $L^1$. Almost-sure convergence and uniform integrability therefore give

We next show that $X_n$ converges weakly to $X$ in the L2 space. For $Y\in L^2$, let $Y_M=Y{1}_{\{|Y|\le M\}}$. Thenwhile the Cauchy-Schwarz inequality gives, uniformly in $n$,Letting first $n\to \infty$ and then $M\to \infty$ proves $\mathbb{E}[X_{n}Y]\to \mathbb{E}[XY]$. Taking $Y=X$ and expanding the square now givesConsequentlyThis proves the almost-sure convergence and convergence of second moments criterion.