0.1 Uniform Integrability
Uniform integrability is a compactness type concept for families of random variables, not unlike that of tightness. Uniform integrability is an extension to the notion of a family of functions being dominated in \(L_{1}\) which is central in dominated convergence.
[!definition] Uniform Integrability Given an arbitrary set \(\mathcal{T}\), a family \(\{ X_{t},~t \in\mathcal{T} \}\) of random variables is uniformly integrable if \[\lim_{x \to \infty}\sup_{t \in\mathcal{T}}\mathbb{E}[|X_{t}|\cdot \mathbb{1}_{|X_{t}|\geq x}]=0.\]
Recall: We say that \(X\) is integrable if and only if \(\mathbb{E}|X|<\infty\) or equivalently, if and only if \(\mathbb{E}[\lvert X_{n} \rvert\mathbb{1}_{\{ \lvert X_{n}\geq M \rvert \}}]\to 0\) as \(M \to \infty\).
We often discuss the case \(\mathcal{T}:=\mathbb{N}_{0}\) and we may define it for an arbitrary index set.
Remark: It follows from the dominated convergence theorem that \[ \lim_{x \to \infty}\mathbb{E}[|X_{t}|\mathbb{1}_{|X_{t}|\geq x}]=0\quad\iff X_{t}\in L^1, \] i.e. that for integrable random variables, far tails contribute little to the expectation. Uniformly integrable families are simply those for which the size of this contribution can be controlled uniformly over all elements.
0.2 Sufficient Conditions for Uniform Integrability
The following are sufficient conditions for uniform integrability of \((X_{n})\):
If for all \(n\), \(|X_{n}|\leq Y\) for some integrable RV \(Y\) (i.e. \(\mathbb{E}|Y|<\infty\)), then the sequence \((X_{n})\) is uniform integrable.
Let \(\phi \geq 0\) be any non-negative function on \([0,\infty)\) such that \(\frac{\phi(x)}{x}\nearrow \infty\) as \(x \to \infty\). If for all \(n\), \(\mathbb{E}[\phi(|X_{n}|)]\leq C\) for some \(C>0\), then sequence \((X_{n})\) is uniformly integrable.
Suppose there exists \(\delta>0\) such that \(sup_{n}\mathbb{E}|X_{n}|^{1+\delta}<\infty\), then the sequence \((X_{n})\) is uniformly integrable.
\begin{proof} See [[files-notes-pstat213bc-lecture-notes-raya-pdf|Raya Feldman’s PSTAT213BC Lecture Notes]].\end{proof}
The conditions above are sufficient but the lemma below gives the necessary and sufficient conditions for uniform integrability.
The sequence \((X_{n})\) is uniformly integrable if and only if the following two conditions hold: 1. \(\sup_{n}\mathbb{E}[|X_{n}|]<\infty\); and 2. \(\forall\varepsilon>0,~\exists \delta>0~s.t.\forall n,~\mathbb{E}[|X_{n}|\mathbb{1}_{A}]<\varepsilon\) for any event \(A\) such that \(\mathbb{P}(A)<\delta\).
\begin{proof} See [[files-notes-pstat213bc-lecture-notes-raya-pdf|Raya Feldman’s PSTAT213BC Lecture Notes]].\end{proof}
The interpretation of this result is that the sequence \((X_{n})\) is uniformly integrable iff and only if its first moments are uniformly bounded and \(X_{n}\)’s cannot take large values on sets of small probability.
0.3 Uniform Integrability Results
For \(p>0\) assume that \(X_n \in L^p\), \(n \geq 1\) and \(X_{n}\stackrel{\mathbb{P}}{\to} X\). The following three statements are equivalent: 1. \(\{ |X_{n}|^p,~n\geq 1 \}\) is uniformly integrable; 2. \(X_{n}\stackrel{L^p}{\to}X\) and \(X\in L^p\); and 3. \(\lim_{n \to \infty}\mathbb{E}[|X_{n}|^p]=\mathbb{E}[|X|^p]<\infty\).
0.4 Uniform Integrability Examples
0.4.0.1 Resources
- Lecture 12: Uniform Integrability, Gordan Zitkovic, Fall 2015 - [[uniform-integrability-zitkovic-2015-pdf]].
- Uniform Integrability, Karl Sigman 2009 - [[uniform-integrability-pdf]]