The bi-Poisson process: a quadratic harness

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.Comment: 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

Renorming divergent perpetuities

We consider a sequence of random variables $(R_n)$ defined by the recurrence $R_n=Q_n+M_nR_{n-1}$, $n\ge1$, where $R_0$ is arbitrary and $(Q_n,M_n)$, $n\ge1$, are i.i.d. copies of a two-dimensional random vector $(Q,M)$, and $(Q_n,M_n)$ is independent of $R_{n-1}$. It is well known that if $E{\ln}|M|<0$ and $E{\ln^+}|Q|<\infty$, then the sequence $(R_n)$ converges in distribution to a random variable $R$ given by $R\stackrel{d}{=}\sum_{k=1}^{\infty}Q_k\prod_{j=1}^{k-1}M_j$, and usually referred to as perpetuity. In this paper we consider a situation in which the sequence $(R_n)$ itself does not converge. We assume that $E{\ln}|M|$ exists but that it is non-negative and we ask if in this situation the sequence $(R_n)$, after suitable normalization, converges in distribution to a non-degenerate limit.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ297 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Infinitesimal generators for a family of polynomial processes -- an algebraic approach

Quadratic harnesses are time-inhomogeneous Markov polynomial processes with linear conditional expectations and quadratic conditional variances with respect to the past-future filtrations. Typically they are determined by five numerical constants hidden in the form of conditional variances. In this paper we derive infinitesimal generators of such processes, extending previously known results. The infinitesimal generators are identified through a solution of a q-commutation equation in the algebra Q of infinite sequences of polynomials in one variable. The solution is a special element in Q, whose coordinates satisfy a three-term recurrence and thus define a system of orthogonal polynomials. It turns out that the respective orthogonality measure uniquely determines the infinitesimal generator (acting on polynomials or bounded functions with bounded continuous second derivative) as an integro-differential operator with the explicit kernel, where the integration is with respect to this measure