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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

The following definition applies.

• A class ${\mathcal {C}}$ of random variables is called uniformly integrable (UI) if given $\epsilon >0$, there exists $K\in [0,\infty )$ such that $E(|X|I_{|X|\geq K})\leq \epsilon \ {\text{ for all X}}\in {\mathcal {C}}$, 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 ${\mathcal {C}}$ 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

The following results apply.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

• 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 $\Omega =\mathbb {R}$, 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,

$E(|X_{n}|,|X_{n}|\geq K)=1\ {\text{ for all }}n\geq K,$

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 not satisfied as the $X_{n}$s are not bounded in $L^{1}$. 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 $L^{1}$. It can also be shown that any $L^{1}$ random variable will satisfy clause 2 in Definition 2.

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

$\ |X_{n}(\omega )|\leq |Y(\omega )|,\ Y(\omega )\geq 0,\ E(Y)<\infty ,$

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

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

The family $\{X_{\alpha }\}_{\alpha \in \mathrm {A} }\subset L^{1}(\mu )$ 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

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• 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. This is a generalization of the dominated convergence theorem.

Citations

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References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• J. Diestel and J. Uhl (1977). Vector measures, Mathematical Surveys 15, American Mathematical Society, Providence, RI ISBN 978-0-8218-1515-1