Uniform integrability
From Wikipedia, the free encyclopedia
In probability theory, the family 
is said to be uniformly integrable if
![\sup_{\alpha}\mathrm{E}\left[ |X_{\alpha}| I_{\{|X_{\alpha}| > c\}} \right]\to 0,\; c\to\infty](../../../../math/f/8/9/f89ac80b2d87acbeb89cec17ebf1b75e.png)
.
This definition is useful in limit theorems, such as Lévy's convergence theorem.
[edit] Sufficient conditions
- Clearly, if
then the family is uniformly integrable.
- The family is uniformly integrable iff it is uniformly bounded (i.e.
) and absolutely continuous (i.e. ![\sup_{\alpha} \mathrm{E} \left[ |X_{\alpha}|I_A\right]\to 0](../../../../math/2/5/c/25cb5de19ee70701786f9aa11a6c0014.png)
as 
).
- (Vallée-Poussin) The family is uniformly integrable iff there exists a nonnegative increasing function G(t) such that
and 
[edit] References
- A.N.Shiryaev (1995). Probability, 2nd Edition, Springer-Verlag, New York, pp.187-188, ISBN 978-0387945491
