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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.
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