Equivalence of equicontinuity concepts for Markov operators derived from a Schur-like property for spaces of measures

Various equicontinuity properties for families of Markov operators have been – and still are – used in the study of existence and uniqueness of invariant probability for these operators, and of asymptotic stability. We prove a general result on equivalence of equicontinuity concepts. It allows compa...

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Bibliographic Details
Published inStatistics & probability letters Vol. 169; p. 108964
Main Authors Hille, Sander C., Szarek, Tomasz, Worm, Daniel T.H., Ziemlańska, Maria A.
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.02.2021
Subjects
Dual bounded Lipschitz norm
Equicontinuity
Markov operators
Measure
Schur property
Weak topology
37A30
Equicontinuity
Schur property
Dual bounded Lipschitz norm
Markov operators
60B10
Measure
Weak topology
46E27
46N30
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Summary:Various equicontinuity properties for families of Markov operators have been – and still are – used in the study of existence and uniqueness of invariant probability for these operators, and of asymptotic stability. We prove a general result on equivalence of equicontinuity concepts. It allows comparing results in the literature and switching from one view on equicontinuity to another, which is technically convenient in proofs. More precisely, the characterisation is based on a ‘Schur-like property’ for measures: if a sequence of finite signed Borel measures on a Polish space is such that it is bounded in total variation norm and such that for each bounded Lipschitz function the sequence of integrals of this function with respect to these measures converges, then the sequence converges in dual bounded Lipschitz norm to a measure.
ISSN:0167-7152
1879-2103
DOI:10.1016/j.spl.2020.108964