Uniform Integrability Criteria

Bounded families are UI

If there exists $B>0$ such that

$$ |X_n|\le B \qquad \text{for all } n, $$

then the family is uniformly integrable.

If for some $p>1$,

$$ \sup_{n\ge 0}\mathbb E[|X_n|^p]<\infty, $$

then $(X_n)$ is uniformly integrable.

The usual estimate is

$$ \mathbb E[|X_n|\mathbf 1(|X_n|\ge K)] \le \frac{\mathbb E[|X_n|^p]}{K^{p-1}}. $$

$$ X_n=\mathbb E[X\mid \mathcal F_n] $$

for some integrable random variable $X$, then $(X_n)$ is automatically uniformly integrable.

If $X_n\to X_\infty$ almost surely but

$$ \mathbb E[X_\infty]\ne \lim_{n\to\infty}\mathbb E[X_n], $$

then the family cannot be uniformly integrable.