Uniform integrability

In probability theory, the family $$\{X_{\alpha}\}_{\alpha\in\Alpha}$$ is said to be uniformly integrable if
 * $$\sup_{\alpha}\mathrm{E}\left[ |X_{\alpha}| I_{\{|X_{\alpha}| > c\}} \right]\to 0,\; c\to\infty$$.

This definition is useful in limit theorems, such as Lévy's convergence theorem.

Sufficient conditions

 * Clearly, if $$\forall\alpha\; |X_{\alpha}| \le \eta,\; \mathrm{E}\eta < \infty$$ then the family $$\{X_{\alpha}\}_{\alpha\in\Alpha}$$ is uniformly integrable.


 * The family $$\{X_{\alpha}\}_{\alpha\in\Alpha}$$ is uniformly integrable iff it is uniformly bounded (i.e. $$\sup_{\alpha}E(|X_{\alpha}|)<\infty$$) and absolutely continuous (i.e. $$\sup_{\alpha} \mathrm{E} \left[ |X_{\alpha}|I_A\right]\to 0$$ as $$\mathrm{P}(A)\to 0$$).


 * (Vallée-Poussin) The family $$\{X_{\alpha}\}_{\alpha\in\Alpha}$$ is uniformly integrable iff there exists a nonnegative increasing function $$G(t)$$ such that $$\lim_t \frac{G(t)}{t} = \infty$$ and $$\sup_{\alpha} E(G(|X_{\alpha}|)) < \infty$$