In the paper we propose conditions ensuring the central limit theorem (CLT) and the law of the iterated logarithm (LIL) for a certain class of Markov chains. We further use this general criteria to verify the aforementioned limit theorems for a particular disrete-time Markov system. The piecewise-deterministic Markov process defined via interpolation of the explored Markov chain can be used e.g. to describe a model for gene expression. The aim for the future work is to establish the CLT and the LIL for the continuous-time process too.
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