Birth and Death Processes, Random Walks, and Orthogonal Polynomials

Abstract

The study of the time-dependent behavior of birth and death processes (BDP) involves many intricate and interesting orthogonal polynomials, such as Charlier, Meixner, Laguerre, Krawtchouk, and other polynomials from the Askey scheme; other famous orthogonal polynomials not in the Askey scheme could appear also, e.g., orthogonal polynomials related to the Roger-Ramanujan continued fraction [88]. In fact, the three-term recurrence relation lies at the heart of continued fractions, orthogonal polynomials, and birth and death processes. For birth and death processes with complicated birth and death rates, for example, when rates are state dependent or nonlinear, it is almost impossible to find closed form solutions of the transition functions. Due to the difficulties involved in analytical methods, it is pertinent to develop other techniques to gain insight into the behavior of the various system characteristics such as system size probabilities, expected system size, etc. The Karlin and McGregor representation [59], [74] of the transition probabilities, which uses a system of orthogonal polynomials satisfying a three-term recurrence relation involving the birth and death rates, is very useful in understanding the asymptotic behavior of the birth and death process. In fact these polynomials appear in some important distributions, namely, the (doubly) limiting conditional distributions. Also these polynomials play a fundamental role in the study of exponential ergodicity (see, e.g., [33] through [36]).KeywordsRandom WalkOrthogonal PolynomialConditional DistributionDeath ProcessKrawtchouk PolynomialThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.