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 ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} be a probability space and ξ 1 , ξ 2 , … {\displaystyle \xi _{1},\xi _{2},\dots } be a sequence of real-valued random variables defined on this space with sup n E [ | ξ n | ] < ∞ . {\displaystyle \sup \limits _{n}\mathbb {E} [|\xi _{n}|]<\infty .}
Then there exists a random variable ψ ∈ L 1 ( P ) {\displaystyle \psi \in L^{1}(P)} and a subsequence ( η k ) = ( ξ n k ) {\displaystyle (\eta _{k})=(\xi _{n_{k}})} , such that for every arbitrary subsequence ( η ~ n ) = ( η k n ) {\displaystyle ({\tilde {\eta }}_{n})=(\eta _{k_{n}})} when n → ∞ {\displaystyle n\to \infty } then
( η ~ 1 + ⋯ + η ~ n ) n → ψ {\displaystyle {\frac {({\tilde {\eta }}_{1}+\cdots +{\tilde {\eta }}_{n})}{n}}\to \psi }
P {\displaystyle P} -almost surely.
Analytic version Let ( E , A , μ ) {\displaystyle (E,{\mathcal {A}},\mu )} be a finite measure space and f 1 , f 2 , … {\displaystyle f_{1},f_{2},\dots } be a sequence of real-valued functions in L 1 ( μ ) {\displaystyle L^{1}(\mu )} and sup n ∫ E | f n | d μ < ∞ {\displaystyle \sup \limits _{n}\int _{E}|f_{n}|\mathrm {d} \mu <\infty } . Then there exists a function υ ∈ L 1 ( μ ) {\displaystyle \upsilon \in L^{1}(\mu )} and a subsequence ( g k ) = ( f n k ) {\displaystyle (g_{k})=(f_{n_{k}})} such that for every arbitrary subsequence
( g ~ n ) = ( g k n ) {\displaystyle ({\tilde {g}}_{n})=(g_{k_{n}})} if n → ∞ {\displaystyle n\to \infty } then
( g ~ 1 + ⋯ + g ~ n ) n → υ {\displaystyle {\frac {({\tilde {g}}_{1}+\cdots +{\tilde {g}}_{n})}{n}}\to \upsilon }
μ {\displaystyle \mu } -almost everywhere.
Explanations So the theorem says, that the sequence ( η k ) {\displaystyle (\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.
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