Uniform integrability

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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.

[edit] 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

[edit] References

  • A.N.Shiryaev (1995). Probability, 2nd Edition, Springer-Verlag, New York, pp.187-188, ISBN 978-0387945491
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