The bi-Poisson process: a quadratic harness

Abstract

This paper is a continuation of our previous research on quadratic harnesses, that is, processes with linear regressions and quadratic conditional variances. Our main result is a construction of a Markov process from given orthogonal and martingale polynomials. The construction uses a two-parameter extension of the Al-Salam--Chihara polynomials and a relation between these polynomials for different values of parameters.

Published in at http://dx.doi.org/10.1214/009117907000000268 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• $q$-Gaussian processes: non-commutative and classical aspects
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• Fock space associated to Coxeter group of type B
• $q-$Wiener ($\alpha ,q)-$ Ornstein-Uhlenbeck processes. A generalization of known processes
• Befriending Askey-Wilson polynomials
• Askey-Wilson signed measures and open ASEP in the shock region
• A few remarks on quadratic harnesses
• Markov processes, polynomial martingales and orthogonal polynomials
• On the families of polynomials forming a part of the Askey--Wilson scheme and their probabilistic applications
• Rejoinder: Gibbs Sampling, Exponential Families and Orthogonal Polynomials
• Stationary, Markov, stochastic processes with polynomial conditional moments and continuous paths
• Hypergroups and Quantum Bessel Processes of Non-integer Dimensions