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.

Komlós' theorem

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.