Uniformly integrable set of random variables

A set of random variables (cf. Random variable) having finite expectations such that integrated tails of their distribution functions are uniformly small. Let a set $ {\mathcal X} $ consist of random variables defined on a common probability space $ ( \Omega, {\mathcal F}, {\mathsf P} ) $. It is called uniformly integrable if

$$ {\lim\limits } _ {c \rightarrow \infty } \sup _ {X \in {\mathcal X} } {\mathsf E} ( X; \left | X \right | > c ) = $$

$$ = {\lim\limits } _ {c \rightarrow \infty } \sup _ {X \in {\mathcal X} } \int\limits _ {\left \{ \omega : {\left | {X ( \omega ) } \right | > c } \right \} } X ( \omega ) { {\mathsf P} ( d \omega ) } = 0 . $$

Uniform integrability is a kind of compactness of sets of random variables or their distribution functions. It plays a key role in a variety of convergence problems. An example of this is the following theorem [a1].

Theorem 1.

Let a sequence $ {\mathcal X} = \{ X _ {n} \} _ {n \geq 0 } $ of random variables such that $ {\mathsf E} | {X _ {n} } | < \infty $, $ n \geq 0 $, converge in probability to a random variable $ X $( cf. Convergence in probability). Then $ {\mathsf E} | X | < \infty $ and $ {\lim\limits } _ {n \rightarrow \infty } {\mathsf E} | {X _ {n} - X } | = 0 $ if and only if the set $ {\mathcal X} $ is uniformly integrable.

In fact, the definition of uniform integrability is stated in terms of marginal distribution functions of random variables $ X \in {\mathcal X} $ and does not necessarily require that all these random variables are defined on the same probability space.

Each finite set of random variables having finite absolute expectations is uniformly integrable. This does not hold, in general, for infinite sets.

Theorem 2.

A set $ {\mathcal X} $ of random variables is uniformly integrable if and only if there exists a non-negative increasing convex function $ G : {\mathbf R _ {+} } \rightarrow {\mathbf R _ {+} } $ such that

$$ {\lim\limits } _ {t \rightarrow \infty } { \frac{G ( t ) }{t} } = \infty $$

and

$$ \sup _ {X \in {\mathcal X} } {\mathsf E} G ( \left | X \right | ) = g < \infty . $$

The criterion above leads to a quantification of the notion of uniform integrability: The straightforward estimate

$$ \sup _ {X \in {\mathcal X} } {\mathsf E} ( \left | X \right | ; \left | X \right | > c ) \leq { \frac{c ( g - G ( 0 ) ) }{G ( c ) - G ( 0 ) } } $$

represents a uniform upper bound of the integrated tails of all random variables belonging to a uniformly integrable set.

References

[a1]	P.A. Meyer, "Probability and potentials" , Blaisdell (1966)