Author(s): Anuradha

Abstract: Let {X_j:j≥-m+1} be a homogeneous Markov chain of order m taking values in {0,1}. For j=0,-1,…,-l+1, we will set R_j=0 and we define R_j=∏_(i=j-1)^(j-l) (1- ├ R_i ) ∏_(i=j)^(j+k-1) X_i. Now R_j=1 implies that an l-look-back run of length k has occured starting at j. Here R_j is defined inductively as a run of 1 's starting at j, provided that no l-look-back run of length k occurs, starting at time j-1,j-2,…,j-l respectively. We study the conditional distribution of the number of l_1-look-back runs of length k_1 until the stopping time i.e. the r-th occurrence of the l-look-back run of length k where k_1≤k and obtain it's probability generating function. The number of l_1-look-back runs of length k_1 until the stopping time has been expressed as the sum of r independent random variables with the first random variable having a slightly different distribution under certain conditions.

DOI: 10.22271/maths.2023.v8.i1a.930

Pages: 04-11 | Views: 237 | Downloads: 15

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