On waiting time problems associated with runs in Markov dependent trials
type: Item
instance of: scholarly article
title: On waiting time problems associated with runs in Markov dependent trials
zbMATH Open document ID: 0949.60085
DOI: 10.1023/A:1003862225719
author: Demetrios L. Antzoulakos
published in: Annals of the Institute of Statistical Mathematics
publication date: 2000-12-03T00:00:00Z
review text: The aim of the paper is to develop a general technique useful to the study of waiting time distribution for the \(k\)th occurrence of a success run of length \(k\) in a sequence of Markov dependent trials. Let \(W^{(a)}_{r,k}\) be a random variable denoting the number of Markov dependent trials until the occurrence of the \(r\)th success run of length \(k\). The superscript a denotes the counting scheme employed and takes the following values: \(a=\text{I}\) for the non-overlapping scheme; \(a=\text{II}\) for the at least \(k\) scheme; \(a=\text{III}\) for the overlapping scheme; \(a=\text{IV}\) for the exactly \(k\) scheme. Also, let \(W_S\) \((W_L)\) be a random variable denoting the waiting time for a run of \(k\) successes or (and) a run of \(s\) failures whichever comes sooner (later). The proposed technique for the study of \(W^{(a)}_{r,k}\), \(W_S\), and \(W_L\) is equivalent to the method of generalized probability generating functions. Section 2 provides the necessary tools for the study of the waiting time distributions. In Section 3, by exploiting the Markov chain embedding technique, the probability mass function and the probability generating function are derived for \(W^{(a)}_{r,k}\), \(W_S\), and \(W_L\).
reviewed by: Neculai Curteanu
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 60J20
Mathematics Subject Classification ID: 60E10
zbMATH DE Number: 1368793
zbMATH Keywords: probability mass function
zbMATH Keywords: probability generating function
zbMATH Keywords: waiting time
zbMATH Keywords: run
zbMATH Keywords: sooner and later problems
zbMATH Keywords: distributions of order \(k\)
MaRDI profile type: Publication
full work available at URL: a:1003862225719
OpenAlex ID: W2070869593
description: scientific article; zbMATH DE number 1368793