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.

The Law of the Iterated Logarithm for Random Dynamical System with Jumps and State-Dependent Jump Intensity

“…Limit theorems for Markov processes have recently been the subject of intense research (see e.g. [1,6,9,10,16]). The LIL defines a range in which, with probability 1, from a certain point the trajectories of the stochastic process will be found.…”