Abstract
We consider a {0,1}-valuedm-th order stationary Markov chain. We study the occurrences of runs where two 1’s are separated byat most/exactly/at least k 0’s under the overlapping enumeration scheme wherek≥0 and occurrences of scans (at leastk 1 successes in a window of length at mostk, 1≤k 1≤k) under both non-overlapping and overlapping enumeration schemes. We derive the generating function of first two types of runs. Under the conditions, (1) strong tendency towards success and (2) strong tendency towards reversing the state, we establish the convergence of waiting times of ther-th occurrence of runs and scans to Poisson type distributions. We establish the central limit theorem and law of the iterated logarithm for the number of runs and scans up to timen.
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Sarkar, A., Sen, K. & Anuradha Waiting time distributions of runs in higher order Markov chains. Ann Inst Stat Math 56, 317–349 (2004). https://doi.org/10.1007/BF02530548
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DOI: https://doi.org/10.1007/BF02530548