Komlós' theorem

Template:Short description 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.

Komlós' theorem

Probabilistic version

Let $(\mathrm{\Omega },ℱ,P)$ be a probability space and ${\xi }_1,{\xi }_2,\dots$ be a sequence of real-valued random variables defined on this space with $\sup {\mathrm{\backslash limits}}_n𝔼[|{\xi }_n|]<\mathrm{\infty }.$

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 $n\to \mathrm{\infty }$ then

$\frac{({\tilde{\eta }}_1+\cdots +{\tilde{\eta }}_n)}{n}\to \psi$

$P$-almost surely.

Analytic version

Let $(E,𝒜,\mu )$ be a finite measure space and $f_1,f_2,\dots$ be a sequence of real-valued functions in $L^1(\mu )$ and $\sup {\mathrm{\backslash limits}}_n\underset{E}{\int }|f_n|\mathrm{d}\mu <\mathrm{\infty }$. 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 $n\to \mathrm{\infty }$ 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.