Uniform integrability

Uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales.

Definition

Let $(X,\mathfrak{M}, \mu)$ be a positive measure space. A set $\Phi\subset L^1(\mu)$ is called uniformly integrable if to each $\epsilon>0$ there corresponds a $\delta>0$ such that

$\left| \int_E f d\mu \right| < \epsilon$

whenever $f \in \Phi$ and $\mu(E)<\delta.$

Formal definition

The following definition applies.^[1]

A class $\mathcal{C}$ of random variables is called uniformly integrable (UI) if given $\epsilon>0$ , there exists $K\in[0,\infty)$ such that $E(|X|I_{|X|\geq K})\le\epsilon\ \text{ for all X} \in \mathcal{C}$ , where $I_{|X|\geq K}$ is the indicator function $I_{|X|\geq K} = \begin{cases} 1 &\text{if } |X|\geq K, \\ 0 &\text{if } |X| < K. \end{cases}$ .

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 $K$ such that, for every $X$ in $\mathcal{C}$ , $\mathrm E(|X|)\leqslant K$ .
- For every $\epsilon > 0$ there exists $\delta > 0$ such that, for every measurable $A$ such that $\mathrm P(A)\leqslant \delta$ and every $X$ in $\mathcal{C}$ , $\mathrm E(|X|:A)\leqslant\epsilon$ .

Related corollaries

The following results apply.

Definition 1 could be rewritten by taking the limits as

\lim_{K \to \infty} \sup_{X \in \mathcal{C}} E(|X|I_{|X|\geq K})=0.

A non-UI sequence. Let $\Omega = [0,1] \subset \mathbb{R}$ , and define

X_n(\omega) = \begin{cases} n, & \omega\in (0,1/n), \\ 0 , & \text{otherwise.} \end{cases}

Clearly

X_n\in L^1

, and indeed

E(|X_n|)=1\ ,

for all n. However,

E(|X_n|,|X_n|\ge K)= 1\ \text{ for all } n\ge K,

and comparing with definition 1, it is seen that the sequence is not uniformly integrable.

Non-UI sequence of RVs. The area under the strip is always equal to 1, but

X_n \to 0

pointwise.

By using Definition 2 in the above example, it can be seen that the first clause is satisfied as $L^1$ norm of all $X_n$ s are 1 i.e., bounded. But the second clause does not hold as given any $\delta$ positive, there is an interval $(0, 1/n)$ with measure less than $\delta$ and $E[|X_m|: (0, 1/n)] =1$ for all $m \ge n$ .
If $X$ is a UI random variable, by splitting

E(|X|)=E(|X|,|X|>K)+E(|X|,|X|<K)

and bounding each of the two, it can be seen that a uniformly integrable random variable is always bounded in

L^1

If any sequence of random variables $X_n$ is dominated by an integrable, non-negative $Y$ : that is, for all ω and n,

\ |X_n(\omega)| \le |Y(\omega)|,\ Y(\omega)\ge 0,\ E(Y)< \infty,

then the class

\mathcal{C}

of random variables

\{X_n\}

is uniformly integrable.

A class of random variables bounded in $L^p$ ( $p>1$ ) is uniformly integrable.

Relevant theorems

Dunford–Pettis theorem^[2]

A class of random variables

X_n \subset L^1(\mu)

is uniformly integrable if and only if it is relatively compact for the weak topology

\sigma(L^1,L^\infty)

de la Vallée-Poussin theorem^[3]

The family

\{X_{\alpha}\}_{\alpha\in\Alpha} \subset L^1(\mu)

is uniformly integrable if and only if there exists a non-negative increasing convex function

G(t)

such that

\lim_{t \to \infty} \frac{G(t)}{t} = \infty

and

\sup_{\alpha} E(G(|X_{\alpha}|)) < \infty.

Relation to convergence of random variables

Main article: convergence of random variables

A sequence $\{X_n\}$ converges to $X$ in the $L_1$ norm if and only if it converges in measure to $X$ 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.^[4] This is a generalization of the dominated convergence theorem.

Citations

↑ 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.

References

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