A Useful Version of the Central Limit Theorem for a General Class of Markov Chains

Abstract

In the paper we propose certain conditions, relatively easy to verify, which ensure the central limit theorem for some general class of Markov chains. To justify the usefulness of our criterion, we further verify it for a particular discrete-time Markov dynamical system. From the application point of view, the examined system provides a useful tool in analysing the stochastic dynamics of gene expression in prokaryotes.

arXiv commentThe new version of the manuscript is focused on the central limit theorem only. The proofs are now more precise. The part concerning the law of the iterated logarithm is postponed to the subsequent paper. Splitting the article into two papers is certainly for the benefit of the reader, since even this single part consists of 25 pages

References in corpus (5)

• Dynamic behavior of stochastic gene expression models in the presence of bursting
• Ergodic properties of some piecewise-deterministic Markov process with application to gene expression modelling
• Exponential rate of convergence for some Markov operators
• Limit Theorems for Some Markov Operators
• Exponential ergodicity of some Markov dynamical system with application to a Poisson driven stochastic differential equation

Cited by in corpus (5)

• The Strassen Invariance Principle for Certain Non-stationary Markov-Feller Chains
• Exponential ergodicity in the bounded-Lipschitz distance for a subclass of piecewise-deterministic Markov processes with random switching between flows
• The central limit theorem for Markov processes that are exponentially ergodic in the bounded-Lipschitz norm
• 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