Komlós' Theorem
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Komlós' theorem is a theorem from
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
and
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
about the Cesàro convergence of a
subsequence In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a ...
of
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s (or
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...
s) and their subsequences to an
integrable In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
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 Finite may refer to: * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Gr ...
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
s. 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,P) be a
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models ...
and \xi_1,\xi_2,\dots be a sequence of real-valued random variables defined on this space with \sup\limits_\mathbb almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur ha ...
.


Analytic version

Let (E,\mathcal,\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\mu<\infty. Then there exists a function \upsilon \in L^1(\mu) and a subsequence (g_k)=(f_) such that for every arbitrary subsequence (\tilde_n)=(g_) if n\to \infty then :\frac{n}\to \upsilon \mu-
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
.


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.


References

Theorems in probability theory Theorems in mathematical analysis>\xi_n, \infty. Then there exists a random variable \psi\in L^1(P) and a subsequence (\eta_k)=(\xi_), such that for every arbitrary subsequence (\tilde_n)=(\eta_) when n\to \infty then :\frac\to \psi P-
almost surely In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur ha ...
.


Analytic version

Let (E,\mathcal,\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\mu<\infty. Then there exists a function \upsilon \in L^1(\mu) and a subsequence (g_k)=(f_) such that for every arbitrary subsequence (\tilde_n)=(g_) if n\to \infty then :\frac{n}\to \upsilon \mu-
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
.


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.


References

Theorems in probability theory Theorems in mathematical analysis