The Strassen Invariance Principle for Certain Non-stationary Markov-Feller Chains

TL;DR

This paper establishes conditions under which a non-stationary Markov-Feller chain satisfies the Strassen invariance principle, using Lyapunov conditions and a specialized coupling, with applications to stochastic gene expression models.

Contribution

It introduces a new criterion combining Lyapunov conditions and a Markovian coupling for the invariance principle in non-stationary Markov chains.

Findings

The criterion applies to certain non-stationary Markov-Feller chains.

It verifies the functional law of the iterated logarithm in complex systems.

An example application to stochastic gene expression dynamics is provided.

Abstract

We propose certain conditions which are sufficient for 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. The construction of such a coupling derives from the paper of M. Hairer (Probab. Theory Related Fields, 124(3):345--380, 2002). Our criterion may serve as a useful tool in verifying the functional law of the iterated logarithm for certain random dynamical systems, developed eg. in molecular biology. In the final part of the paper we present an example application of our main theorem to the mathematical model describing stochastic dynamics of gene expression.