Uniform integrability

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

Formal definition[edit]

The following definition applies.[1]

  • An alternative definition involving two clauses may be presented as follows: A class \mathcal{C} 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[edit]

The following results apply.[citation needed]

  • 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_ns 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[edit]

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).
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[edit]

  • 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[edit]

  1. ^ Williams, David (1997). Probability with Martingales (Repr. ed.). Cambridge: Cambridge Univ. Press. pp. 126–132. ISBN 978-0-521-40605-5. 
  2. ^ Dellacherie, C. and Meyer, P.A. (1978). Probabilities and Potential, North-Holland Pub. Co, N. Y. (Chapter II, Theorem T25).
  3. ^ Meyer, P.A. (1966). Probability and Potentials, Blaisdell Publishing Co, N. Y. (p.19, Theorem T22).
  4. ^ 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[edit]