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

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.

arXiv commentThe results were primary announced in the manuscript arXiv:1804.09220v1, which we finally splitted into two papers in order to be able to give more precise descriptions and proofs. Note that even this single part consists of 30 pages (the other one is 25-pages long)

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• Dynamic behavior of stochastic gene expression models in the presence of bursting
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Cited by in corpus (3)

• Exponential ergodicity in the bounded-Lipschitz distance for a subclass of piecewise-deterministic Markov processes with random switching between flows
• The law of the iterated logarithm for a piecewise deterministic Markov process assured by the properties of the Markov chain given by its post-jump locations
• Continuous dependence of an invariant measure on the jump rate of a piecewise-deterministic Markov process