Uniform Integrability - Related Corollaries

Related Corollaries

The following results apply.

  • Definition 1 could be rewritten by taking the limits as
  • A non-UI sequence. Let, and define
X_n(\omega) = \begin{cases} n, & \omega\in (0,1/n), \\ 0, & \text{otherwise.} \end{cases}
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 not satisfied as the s are not bounded in . 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 . It can also be shown that any random variable will satisfy clause 2 in Definition 2.
  • 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.

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