Uniform Integrability - Related Corollaries

Related Corollaries

The following results apply.

• Definition 1 could be rewritten by taking the limits as
$X_n(omega) = begin{cases} n, & omegain (0,1/n), \ 0, & text{otherwise.} end{cases}$
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.