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Let  X  0 ,  X  1 , … be a simple ergodic Markov chain with a finite set of states and  ξ̃   n, k  ( s ) be a number of runs of  k -fold repetitions of strings having length  s . Estimates of accuracy of the Poisson approximation for the distribution of  ξ   n, k  ( s ) in the sequence  X  0 ,  X  1 , …,  X   n + s −1  are obtained, these estimates are uniform over  k .

Estimates of accuracy of the Poisson approximation for the distribution of number of runs of long string repetitions in a Markov chain