The classical bi-Poisson process: An invertible quadratic harness

We give an elementary construction of a time-invertible Markov process which is discrete except at one instance. The process is one of the quadratic harnesses studied in Bryc and Wesołowski [2005. Conditional moments of q-Meixner processes. Probab. Theory Related Fields 131, 415-441 〈arxiv.org/abs/math.PR/0403016〉], Bryc et al. [2005b. Quadratic harnesses, q-commutations, and orthogonal martingale polynomials. Trans. Amer. Math. Soc. 〈arxiv.org/abs/math.PR/0504194〉, to appear], and Bryc et al. [2005a. The bi-Poisson process: a quadratic harness 〈arxiv.org/abs/math.PR/0510208〉]. It can be constructed from a pair of independent Poisson processes with the same gamma-distributed intensity.