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Uniform integrability

If $\sup_n\mathbb E[X_n^2]<\infty$, then $(X_n)$ is <uniform integrability>[uniformly integrable]. Indeed, the <Cauchy-Schwarz inequality> and <Markov inequality> give
$$
\mathbb E[|X_n|\mathbf1_{\{|X_n|>K\}}]
\leq\frac{\mathbb E[X_n^2]}{K}.
$$

Uniform integrability from bounded second moments

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 202 / 3 / c / Solution

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

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

Uniform integrability from bounded second moments

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Uniform integrability from bounded second moments

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