Cumulative Bernoulli trials and Krawtchouk processes
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.
scientific article; zbMATH DE number 3848350