What is it about?
Certain Markov chains which evolve on a general metric space (that is a Polish space), are investigated in terms of their ergodic properties. In this publication the law of the iterated logarithm is established for them. In fact, we examine not the chains themselves, but the associated Markov operators (acting on measures). Our previous paper [Limit theorems for some Markov chains, J. Math. Anal. Appl., 443 (1), 385-408 (2016)] concerns, among others, asymptotic stability for the chosen class of Markov operators, so the existence and the uniqueness of measures which are invariant to these operators, as well as the convergence (in the Fortet-Mourier norm) to these measures. Moreover, the rate of convergence is estimated to be exponential, which allows for the verification of the limit theorems (also the law of the iterated logarithm, proven in this paper). Our proofs require the use of the so-called Markovian coupling (introduced by M. Hairer for the chains with infinite state space), creatively combined with the martingale method.
Why is it important?
The Markov chains investigated within this paper correspond to the iterated function systems applied to biological models for cell cycle and gene expression, see e.g. [S.C. Hille, K. Horbacz, T. Szarek, Existence of a unique invariant measure for a class of equicontinuous Markov operators with application to a stochastic model for an autoregulated gene. Ann. Math. Blaise Pascal 23(2), 171–217 (2016)], [A. Lasota, M.C. Mackey, Cell division and the stability of cellular populations, J. Math. Biol. 38, 241–261 (1999)], [J.J. Tyson, K.B. Hannsgen, Cell growth and division: a deterministic/ probabilistic model of the cell cycle, J. Math. Biol. (23), 231–246 (1986)].