Cumulative Bernoulli trials and Krawtchouk processes
type: Item
instance of: scholarly article
title: Cumulative Bernoulli trials and Krawtchouk processes
zbMATH Open document ID: 0534.60060
DOI: 10.1016/0304-4149(84)90014-0
author: Mizan Rahman
author: Michael R. Hoare
published in: Stochastic Processes and their Applications
publication date: 1984-01-01T00:00:00Z
review text: The authors denote by cumulative Bernoulli trials (CBT's) a family of Markov chains associated with combined Bernoulli trials allowing ''saved'' and ''recouped'' successes. The simplest such process, well known in gambling, consists of saving successes in a sequence of throws with N, say, dice. The main object of the paper is the following variant (called the Krawtchouk process): Let a player with N dice throw only a selection i of them with success probability \(\alpha\). Suppose he ''puts aside'' the k successes and throws the N-k remaining dice with success probability \(\beta\), thereby obtaining \(\ell\) additional successes. The process is repeated with i (the initial state) replaced by \(k+\ell\) (the state at ''time'' 1). The transition probabilities of the resulting Markov chain are easy to write down. Possible applications of the model in epidemic and reliability theory are described. The authors compute the stationary distribution, the moments, the autocorrelation function and the eigenvalues of the transition matrix K. The problem of obtaining the eigenvectors of K is considered next. As demonstrated, these are given in the form of Krawtchouk polynomials (the set orthogonal on the integers \(i\in \{0,1,...,N\}\) with binomial weight). The authors consider also modifications of the Krawtchouk process obtained when (i) \(N\to \infty\), \(\beta\to 0\), \(N\beta =\mu\) (so that the second ''throw'' is replaced by a Poisson experiment) and (ii) the Bernoulli trials are replaced by suitably defined hypergeometric trials. The eigenvectors associated with the modifications (i) and (ii) are given by polynomials known as Charlier and Gonin polynomials, respectively.
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 60K10
Mathematics Subject Classification ID: 60K99
zbMATH DE Number: 3848350
zbMATH Keywords: Bernoulli trials
zbMATH Keywords: Krawtchouk process
zbMATH Keywords: reliability theory
zbMATH Keywords: stationary distribution
reviewed by: Bo Henry Lindqvist
MaRDI profile type: Publication
full work available at URL: 0304-4149(84)90014-0
OpenAlex ID: W1990820285
cites work: XI.—On Fourfold Sampling with and without Replacement
cites work: Q4170993
cites work: Stochastic processes and special functions: On the probabilistic origin of some positive kernels associate with classical orthogonal polynomials
cites work: Q5528194
cites work: Q5773713
cites work: Q5820721
cites work: Q3241504
cites work: The use of orthogonal polynomials of the positive and negative binomial frequency functions in curve fitting by Aitken's method
cites work: Random Walk and the Theory of Brownian Motion
cites work: Q5341273
cites work: Q5556830
cites work: Some Positive Kernels and Bilinear Sums for Hahn Polynomials
cites work: Q3267566
cites work: Q4099514
Wikidata QID: Q128066854
description: scientific article; zbMATH DE number 3848350