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Title: Weak law of large numbers for iterates of random-valued functions
Authors: Baron, Karol
Keywords: Random-valued functions; Iterates; Weak law of large numbers; Convergence in law; Convergence in probability
Issue Date: 2019
Citation: Aequationes Mathematicae, Vol. 93, no. 2 (2019), s. 415-423
Abstract: Given a probability space (Ω,A, P), a complete and separable metric space X with the σ-algebra B of all its Borel subsets and a B⊗A-measurable f : X ×Ω → X we consider its iterates fn defined on X × ΩN by f0(x, ω) = x and fn(x, ω) = f fn−1(x, ω), ωn for n ∈ N and provide a simple criterion for the existence of a probability Borel measure π on X such that for every x ∈ X and for every Lipschitz and bounded ψ : X → R the sequence 1 n n−1 k=0 ψ fk(x, ·) n∈N converges in probability to X ψ(y)π(dy).
DOI: 10.1007/s00010-018-0585-0
ISSN: 1420-8903
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