Uniform integrability
From Wikipedia, the free encyclopedia
In probability theory, the family is said to be uniformly integrable if
.
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.
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