Komlós' theorem explained

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.^[1] There exists also a generalization from 1970 by Srishti D. Chatterji.^[2]

Komlós' theorem

Probabilistic version

Let

(\Omega, l{F},P) be a probability space and \xi _{1,\xi} _{2,...} be a sequence of real-valued random variables defined on this space with \sup\limits _{n} E [|\xi _{n|]<infty.}

Then there exists a random variable

\psi\in L ^{1(P)} and a subsequence ( η _{k)=(\xi} ^{n _{k}} ) , such that for every arbitrary subsequence (\tilde{ η } _{n)=(η} ^{k _{n}} ) when n\to inf ty then((\tilde{ η)/(_{1+ …}) + \tilde{ η } _{n)}{n}\to} \psi P -almost surely.

Analytic version

Let

(E, l{A},\mu) be a finite measure space and f _{1,f} _{2,...} be a sequence of real-valued functions in L ^{1(\mu)} and \sup\limits _{n} \int _{E} |f _{n|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 inf ty then((\tilde{g)/(_{1+ …}) + \tilde{g} _{n)}{n}\to} \upsilon \mu -almost everywhere.

Explanations

So the theorem says, that the sequence

( η _{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.

References

1. János Komlós. 1967. 10.1007/BF02020976. 1. Acta Mathematica Academiae Scientiarum Hungaricae. A Generalisation of a Problem of Steinhaus. 18.
2. S. D. Chatterji. 1970. 10.1007/BF01404326. 235–245. Inventiones Mathematicae. A general strong law. 9.