Weak law of large numbers for iterates of random-valued functions

Baron, Karol

doi:10.1007/s00010-018-0585-0

Cited by 5 publications

(4 citation statements)

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“…for x ∈ X and for any continuous and bounded u : X → R; moreover, as observed in [3] (see also [6]),…”

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confidence: 79%

Remarks Connected with the Weak Limit of Iterates of Some Random-Valued Functions and Iterative Functional Equations

Baron

2020

Annales Mathematicae Silesianae

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The paper consists of two parts. At first, assuming that (Ω, 𝒜, P) is a probability space and (X, ϱ) is a complete and separable metric space with the σ-algebra 𝒝 of all its Borel subsets we consider the set 𝒭c of all 𝒝 ⊗ 𝒜-measurable and contractive in mean functions f : X × Ω → X with finite integral ∫ Ω ϱ (f(x, ω), x) P (dω) for x ∈ X, the weak limit πf of the sequence of iterates of f ∈ 𝒭c, and investigate continuity-like property of the function f ↦πf, f ∈ 𝒭c, and Lipschitz solutions φ that take values in a separable Banach space of the equation \varphi \left( x \right) = \int_\Omega {\varphi \left( {f\left( {x,\omega } \right)} \right)P\left( {d\omega } \right)} + F\left( x \right).Next, assuming that X is a real separable Hilbert space, Λ: X → X is linear and continuous with ||Λ|| < 1, and µ is a probability Borel measure on X with finite first moment we examine continuous at zero solutions φ : X → 𝔺 of the equation \varphi \left( x \right) = \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \mu } \left( x \right)\varphi \left( {\Lambda x} \right) which characterizes the limit distribution πf for some special f ∈ 𝒭c.

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“…for x ∈ X and for any continuous and bounded u : X → R; moreover, as observed in [3] (see also [6]),…”

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confidence: 79%

Remarks Connected with the Weak Limit of Iterates of Some Random-Valued Functions and Iterative Functional Equations

Baron

2020

Annales Mathematicae Silesianae

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“…A simple criterion for the convergence in law of f n (x, •) n∈N to a random variable independent of x ∈ X was proved in [1], assuming that X is complete and separable. In [2] it has been strengthened and applied to obtain a weak law of large numbers for iterates of random-valued functions. In the present paper we are interested in a strong law of large numbers.…”

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confidence: 99%

Strong Law of Large Numbers for Iterates of Some Random-Valued Functions

Baron

Kapica

2022

Results Math

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Assume $$ (\Omega , {\mathscr {A}}, P) $$ ( Ω , A , P ) is a probability space, X is a compact metric space with the $$ \sigma $$ σ -algebra $$ {\mathscr {B}} $$ B of all its Borel subsets and $$ f: X \times \Omega \rightarrow X $$ f : X × Ω → X is $$ {\mathscr {B}} \otimes {\mathscr {A}} $$ B ⊗ A -measurable and contractive in mean. We consider the sequence of iterates of f defined on $$ X \times \Omega ^{{\mathbb {N}}}$$ X × Ω N by $$f^0(x, \omega ) = x$$ f 0 ( x , ω ) = x and $$ f^n(x, \omega ) = f\big (f^{n-1}(x, \omega ), \omega _n\big )$$ f n ( x , ω ) = f ( f n - 1 ( x , ω ) , ω n ) for $$n \in {\mathbb {N}}$$ n ∈ N , and its weak limit $$\pi $$ π . We show that if $$\psi :X \rightarrow {\mathbb {R}}$$ ψ : X → R is continuous, then for every $$ x \in X $$ x ∈ X the sequence $$\left( \frac{1}{n}\sum _{k=1}^n \psi \big (f^k(x,\cdot )\big )\right) _{n \in {\mathbb {N}}}$$ 1 n ∑ k = 1 n ψ ( f k ( x , · ) ) n ∈ N converges almost surely to $$\int _X\psi d\pi $$ ∫ X ψ d π . In fact, we are focusing on the case where the metric space is complete and separable.

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“…Following an idea from [10] Baron has used the Hutchinson distance of distributions to get the geometric rate of convergence of sequences of distributions of iterates of random valued vector functions in the Fortet-Mourier metric (see [1]). Recently this result has been strengthened in [2] by the fact that the distribution of the limit has the first moment finite. The aim of the present paper, motivated by [1,10], is to show how fast a sequence of iterates of Markov operators tends in the Hutchinson distance to a unique invariant and attractive measure.…”

Section: Introductionmentioning

confidence: 99%

The geometric rate of convergence of random iteration in the Hutchinson distance

Kapica

2018

Aequat. Math.

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Weak law of large numbers for iterates of random-valued functions

Abstract: Given a probability space (Ω, A, P ), a complete and separable metric space X with the σ-algebra

Cited by 5 publications

References 6 publications

Remarks Connected with the Weak Limit of Iterates of Some Random-Valued Functions and Iterative Functional Equations

Remarks Connected with the Weak Limit of Iterates of Some Random-Valued Functions and Iterative Functional Equations

Strong Law of Large Numbers for Iterates of Some Random-Valued Functions

The geometric rate of convergence of random iteration in the Hutchinson distance

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