Continuous-time Markov processes, orthogonal polynomials and Lancaster probabilities

Department of Probability and Statistics, IIMAS, UNAM, Mexico city, Mexico.

^* Corresponding author: fred_25@ciencias.unam.mx

Received: 14 December 2018
Accepted: 15 January 2020

Abstract

This work links the conditional probability structure of Lancaster probabilities to a construction of reversible continuous-time Markov processes. Such a task is achieved by using the spectral expansion of the corresponding transition probabilities in order to introduce a continuous time dependence in the orthogonal representation inherent to Lancaster probabilities. This relationship provides a novel methodology to build continuous-time Markov processes via Lancaster probabilities. Particular cases of well-known models are seen to fall within this approach. As a byproduct, it also unveils new identities associated to well known orthogonal polynomials.