Komlós' theorem

Short description: Theorem

Komlós' theorem is a theorem from probability theory and mathematical analysis about the Cesàro convergence of a subsequence of random variables (or functions) and their subsequences to an integrable random variable (or function). It's also an existence theorem for an integrable random variable (or function). There exist a probabilistic and an analytical version for finite measure spaces.

The theorem was proven in 1967 by János Komlós. There exists also a generalization from 1970 for general measure spaces by Srishti D. Chatterji.

Komlós' theorem

Probabilistic version

Let [math]\displaystyle{ (\Omega,\mathcal{F},P) }[/math] be a probability space and [math]\displaystyle{ \xi_1,\xi_2,\dots }[/math] be a sequence of real-valued random variables defined on this space with [math]\displaystyle{ \sup\limits_{n}\mathbb{E}[|\xi_n|]\lt \infty. }[/math]

Then there exists a random variable [math]\displaystyle{ \psi\in L^1(P) }[/math] and a subsequence [math]\displaystyle{ (\eta_k)=(\xi_{n_{k}}) }[/math], such that for every arbitrary subsequence [math]\displaystyle{ (\tilde{\eta}_n)=(\eta_{k_{n}}) }[/math] when [math]\displaystyle{ n\to \infty }[/math] then

[math]\displaystyle{ \frac{(\tilde{\eta}_1+\cdots +\tilde{\eta}_n)}{n}\to \psi }[/math]

[math]\displaystyle{ P }[/math]-almost surely.

Analytic version

Let [math]\displaystyle{ (E,\mathcal{A},\mu) }[/math] be a finite measure space and [math]\displaystyle{ f_1,f_2,\dots }[/math] be a sequence of real-valued functions in [math]\displaystyle{ L^1(\mu) }[/math] and [math]\displaystyle{ \sup\limits_n \int_E |f_n|\mathrm{d}\mu\lt \infty }[/math]. Then there exists a function [math]\displaystyle{ \upsilon \in L^1(\mu) }[/math] and a subsequence [math]\displaystyle{ (g_k)=(f_{n_{k}}) }[/math] such that for every arbitrary subsequence [math]\displaystyle{ (\tilde{g}_n)=(g_{k_{n}}) }[/math] if [math]\displaystyle{ n\to \infty }[/math] then

[math]\displaystyle{ \frac{(\tilde{g}_1+\cdots +\tilde{g}_n)}{n}\to \upsilon }[/math]

[math]\displaystyle{ \mu }[/math]-almost everywhere.

Explanations

So the theorem says, that the sequence [math]\displaystyle{ (\eta_k) }[/math] and all its subsequences converge in Césaro.

Literature

• Kabanov, Yuri & Pergamenshchikov, Sergei. (2003). Two-scale stochastic systems. Asymptotic analysis and control. 10.1007/978-3-662-13242-5. Page 250.