Komlós' 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 by Srishti D. Chatterji.

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

Then there exists a random variable $$\psi\in L^1(P)$$ and a subsequence $$(\eta_k)=(\xi_{n_{k}})$$, such that for every arbitrary subsequence $$(\tilde{\eta}_n)=(\eta_{k_{n}})$$ when $$n\to \infty$$ then
 * $$\frac{(\tilde{\eta}_1+\cdots +\tilde{\eta}_n)}{n}\to \psi$$

$$P$$-almost surely.

Analytic version
Let $$(E,\mathcal{A},\mu)$$ be a finite measure space and $$f_1,f_2,\dots$$ be a sequence of real-valued functions in $$L^1(\mu)$$ and $$\sup\limits_n \int_E |f_n|\mathrm{d}\mu<\infty$$. Then there exists a function $$\upsilon \in L^1(\mu)$$ and a subsequence $$(g_k)=(f_{n_{k}})$$ such that for every arbitrary subsequence $$(\tilde{g}_n)=(g_{k_{n}})$$ if $$n\to \infty$$ then
 * $$\frac{(\tilde{g}_1+\cdots +\tilde{g}_n)}{n}\to \upsilon $$

$$\mu$$-almost everywhere.

Explanations
So the theorem says, that the sequence $$(\eta_k)$$ 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.