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

TL;DR

This paper establishes the law of the iterated logarithm for a class of piecewise deterministic Markov processes by leveraging ergodic properties of the associated Markov chain at jump times, with applications in biological modeling.

Contribution

It introduces a novel approach to prove the law of the iterated logarithm for continuous-time processes using properties of the discrete-time Markov chain at jump points.

Findings

Law of the iterated logarithm proven for the process

Uses properties of the Markov chain at jump times

Applicable to biological stochastic models

Abstract

In the paper we consider some piecewise deterministic Markov process whose continuous component evolves according to semiflows, which are switched at the jump times of a Poisson process. The associated Markov chain describes the states of this process directly after the jumps. Certain ergodic properties of these two objects have already been investigated in our recent papers. We now aim to establish the law of the iterated logarithm for the aforementioned continuous-time process. Moreover, we intend to do this using the already proven properties of the discrete-time system. The abstract model under consideration has interesting interpretations in real-life sciences, such as biology. Among others, it can be used to describe the stochastic dynamics of gene expression.