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- Title
Continuous dependence of the weak limit of iterates of some random-valued vector functions.
- Authors
Komorek, Dawid
- Abstract
Given a probability space (Ω , A , P) , a complete separable Banach space X with the σ -algebra B (X) of all its Borel subsets, an operator Λ : Ω → L (X , X) and ξ : Ω → X we consider the B (X) ⊗ A -measurable function f : X × Ω → X given by f (x , ω) = Λ (ω) x + ξ (ω) and investigate the continuous dependence of a weak limit π f of the sequence of iterates (f n (x , ·)) n ∈ N of f, defined by f 0 (x , ω) = x , f n + 1 (x , ω) = f (f n (x , ω) , ω n + 1) for x ∈ X and ω = (ω 1 , ω 2 , ⋯) . Moreover for X taken as a Hilbert space we characterize π f via the functional equation φ f (u) = ∫ Ω φ f (Λ (ω) u) φ ξ (u) P (d ω) with the aid of its characteristic function φ f . We also indicate the continuous dependence of a solution of that equation.
- Subjects
VECTOR valued functions; FUNCTIONAL equations; BANACH spaces; BOREL subsets; HILBERT space; QUADRATIC equations
- Publication
Aequationes Mathematicae, 2023, Vol 97, Issue 4, p753
- ISSN
0001-9054
- Publication type
Article
- DOI
10.1007/s00010-023-00959-w