No. 948

Title:

Joint distributions associated with compound patterns in a sequence of Markov dependent multistate trials

Author(s):

Kiyoshi, Inoue (Seikei University);
Sigeo, Aki (Kansai University)

Key words:

Markov chain; multistate trials; patterns; runs; non-overlap counting; overlap counting; probability generating function.

Abstract:

    Let ${Z_t:t \geq 1 }$ be multistate Markov dependent trials and let $\Lambda $ be a compound pattern (a set of $\ell $ distinct simple patterns). In this paper, we study joint distributions of the waiting time until the $r$ th occurrence of the compound pattern $\Lambda $ and the numbers of the each simple pattern observed at that time in the multistate Markov dependent trials. We provide methods for deriving the probability generating functions of the joint distributions under two types of counting schemes (non-overlap counting and overlap counting) for the compound pattern $\Lambda $. Besides, the present work are useful in elucidating the primary difference between non-overlap counting and overlap counting. As applications, when $\Lambda $ is a set of runs, the corresponding joint distributions are investigated and a practical example is mentioned. Finally, we give numerical examples in order to illustrate the feasibility of our results. \par \par