The Strassen invariance principle for certain non-stationary Markov–Feller chains

TitleThe Strassen invariance principle for certain non-stationary Markov–Feller chains
Publication TypeJournal Article
Year of Publication2021
AuthorsCzapla D, Horbacz K, Wojewódka-Ściążko H
JournalAsymptotic Analysis
Volume121
Issue1
Start Page1-34
Date Published12/2020
Keywordsasymptotic coupling, invariant measure, law of the iterated logarithm, Markov chain, random dynamical system
Abstract

We propose certain conditions implying the functional law of the iterated logarithm (the Strassen invariance principle) for some general class of non-stationary Markov–Feller chains. This class may be briefly specified by the following two properties: firstly, the transition operator of the chain under consideration enjoys a non-linear Lyapunov-type condition, and secondly, there exists an appropriate Markovian coupling whose transition probability function can be decomposed into two parts, one of which is contractive and dominant in some sense. Our criterion may serve as a useful tool in verifying the functional law of the iterated logarithm for certain random dynamical systems, developed e.g. in biology and population dynamics. In the final part of the paper we present an example application of our main theorem to a mathematical model describing stochastic dynamics of gene expression.

DOI10.3233/ASY-191592