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- Title
Weak law of large numbers for iterates of random-valued functions.
- Authors
Baron, Karol
- 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 1n∑k=0n-1ψ(fk(x,·))n∈N converges in probability to ∫Xψ(y)π(dy).
- Subjects
LAW of large numbers; PROBABILITY measures; BOREL subsets; METRIC spaces
- Publication
Aequationes Mathematicae, 2019, Vol 93, Issue 2, p415
- ISSN
0001-9054
- Publication type
Article
- DOI
10.1007/s00010-018-0585-0