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

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 dynamical systems have been already 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.

Keywords:

• Asymptotic coupling
• invariant measure
• law of the iterated logarithm
• piecewise deterministic Markov process
• random dynamical system

2010 AMS SUBJECT CLASSIFICATION:

• 60J25
• 60J75
• 60F17
• 47D07

Additional information

Funding

The work of Hanna Wojewódka-Ściążko was supported by the National Science Centre of Poland under Grant 2018/02/X/ST1/01518.