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Let be a positive measure space. A set is called uniformly integrable if to each there corresponds a such that
The following definition applies.
- A class of random variables is called uniformly integrable (UI) if given , there exists such that , where is the indicator function .
- An alternative definition involving two clauses may be presented as follows: A class of random variables is called uniformly integrable if:
- There exists a finite such that, for every in , .
- For every there exists such that, for every measurable such that and every in , .
The following results apply.
- Definition 1 could be rewritten by taking the limits as
- A non-UI sequence. Let , and define
- Clearly , and indeed for all n. However,
- and comparing with definition 1, it is seen that the sequence is not uniformly integrable.
- By using Definition 2 in the above example, it can be seen that the first clause is satisfied as norm of all s are 1 i.e., bounded. But the second clause does not hold as given any positive, there is an interval with measure less than and for all .
- If is a UI random variable, by splitting
- and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in .
- If any sequence of random variables is dominated by an integrable, non-negative : that is, for all ω and n,
- then the class of random variables is uniformly integrable.
- A class of random variables bounded in () is uniformly integrable.
- A class of random variables is uniformly integrable if and only if it is relatively compact for the weak topology .
- The family is uniformly integrable if and only if there exists a non-negative increasing convex function such that
Relation to convergence of random variables
Main article: convergence of random variables
- A sequence converges to in the norm if and only if it converges in measure to and it is uniformly integrable. In probability terms, a sequence of random variables converging in probability also converge in the mean if and only if they are uniformly integrable. This is a generalization of the dominated convergence theorem.
- Williams, David (1997). Probability with Martingales (Repr. ed.). Cambridge: Cambridge Univ. Press. pp. 126–132. ISBN 978-0-521-40605-5.
- Dellacherie, C. and Meyer, P.A. (1978). Probabilities and Potential, North-Holland Pub. Co, N. Y. (Chapter II, Theorem T25).
- Meyer, P.A. (1966). Probability and Potentials, Blaisdell Publishing Co, N. Y. (p.19, Theorem T22).
- Bogachev, Vladimir I. (2007). Measure Theory Volume I. Berlin Heidelberg: Springer-Verlag. p. 268. doi:10.1007/978-3-540-34514-5_4. ISBN 3-540-34513-2.
- A.N. Shiryaev (1995). Probability (2 ed.). New York: Springer-Verlag. pp. 187–188. ISBN 978-0-387-94549-1.
- Walter Rudin (1987). Real and Complex Analysis (3 ed.). Singapore: McGraw–Hill Book Co. p. 133. ISBN 0-07-054234-1.
- J. Diestel and J. Uhl (1977). Vector measures, Mathematical Surveys 15, American Mathematical Society, Providence, RI ISBN 978-0-8218-1515-1