The central limit theorem (CLT) and the law of the iterated logarithm (LIL) are, along the strong law of large numbers (SLLN), the most common limit theorems. Some well-known results concerning limit theorems, obtained mainly due to the martingale method, are gathered in [1]. Although the asymptotic behaviour of stationary and ergodic Markov chains is already well investigated, limit theorems for a wider class of Markov processes are still the subject of research.  Together with D. Czapla and K. Horbacz we have proven certain criteria on the CLT and the LIL for a quite general class of Markov chains. Our aim was to provide useful assertions that can serve biologists and physicists to study their models in terms of limit theorems. Therefore we do not require from the Markov chains to be continuous with respect to their initial conditions (as is necessary to assume for the results in [2,3,4] to hold). We do not even directly require the exponential mixing property (see e.g. [5] for the precise formulation). Instead, we propose certain conditions, relatively easy to verify in many biological models, that yield the exponential ergodicity (according to  [6, Theorem 2.1]), as well as the limit theorems (the CLT and the LIL).  The class of Markov chains for which we establish limit theorems may be shortly specified by the existence of an appropriate Markovian coupling whose transition law can be decomposed into two parts, one of which is contractive and dominant in some sense. The construction of such a coupling is adapted from [6], which, in turn, is  inspired by the prominent results of M. Hairer [5]. To justify the usefulness of stating such criteria, we decided to verify them for a particular discrete-time Markov system, for which we were not able to verify conditions proposed in [2] and [3]. 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.

[1] P. Hall and C.C. Heyde, Martingale limits theory and its applications, Academic Press, New York (1980).

[2] J. Gulgowski, S.C. Hille, T. Szarek, and M. Ziemlańska, Central limit theorem for some non-stationary Markov chains, Submitted (2017).

[3] W. Bołt, A.A. Majewski, and T. Szarek, An invariance principle for the law of the iterated logarithm for some Markov chains, Studia Math, 212:41-53 (2012).

[4] T. Komorowski and A. Walczuk, Central limit theorem for Markov processes with spectral gap in the Wasserstein metric, Stochastic Process. Appl., 122:2155-2184 (2012).

[5] M. Hairer. Exponential mixing properties of stochastic PDEs through asymptotic coupling, Probab. Theory Related Fields, 124(3):345-380 (2002).

[6] R. Kapica and M. Ślęczka. Random iteration with place dependent probabilities, arXiv:1107.0707v3 [math.PR] (2017). 

The central limit theorem (CLT) and the law of the iterated logarithm (LIL) are, along the strong law of large numbers (SLLN), the most common limit theorems. Some well-known results concerning limit theorems, obtained mainly due to the martingale method, are gathered in [1]. Although the asymptotic behaviour of stationary and ergodic Markov chains is already well investigated, limit theorems for a wider class of Markov processes are still the subject of research.  Together with D. Czapla and K. Horbacz we have proven certain criteria on the CLT and the LIL for a quite general class of Markov chains. Our aim was to provide useful assertions that can serve biologists and physicists to study their models in terms of limit theorems. Therefore we do not require from the Markov chains to be continuous with respect to their initial conditions (as is necessary to assume for the results in [2,3,4] to hold). We do not even directly require the exponential mixing property (see e.g. [5] for the precise formulation). Instead, we propose certain conditions, relatively easy to verify in many biological models, that yield the exponential ergodicity (according to  [6, Theorem 2.1]), as well as the limit theorems (the CLT and the LIL).  The class of Markov chains for which we establish limit theorems may be shortly specified by the existence of an appropriate Markovian coupling whose transition law can be decomposed into two parts, one of which is contractive and dominant in some sense. The construction of such a coupling is adapted from [6], which, in turn, is  inspired by the prominent results of M. Hairer [5]. To justify the usefulness of stating such criteria, we decided to verify them for a particular discrete-time Markov system, for which we were not able to verify conditions proposed in [2] and [3]. 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.

[1] P. Hall and C.C. Heyde, Martingale limits theory and its applications, Academic Press, New York (1980).

[2] J. Gulgowski, S.C. Hille, T. Szarek, and M. Ziemlańska, Central limit theorem for some non-stationary Markov chains, Submitted (2017).

[3] W. Bołt, A.A. Majewski, and T. Szarek, An invariance principle for the law of the iterated logarithm for some Markov chains, Studia Math, 212:41-53 (2012).

[4] T. Komorowski and A. Walczuk, Central limit theorem for Markov processes with spectral gap in the Wasserstein metric, Stochastic Process. Appl., 122:2155-2184 (2012).

[5] M. Hairer. Exponential mixing properties of stochastic PDEs through asymptotic coupling, Probab. Theory Related Fields, 124(3):345-380 (2002).

[6] R. Kapica and M. Ślęczka. Random iteration with place dependent probabilities, arXiv:1107.0707v3 [math.PR] (2017).