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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.