Jacek Wesołowski

Conditional moments of q-Meixner processes

Abstract.We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a three-parameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these processes are known to arise from the non-commutative generalizations of the Lévy processes.

Quadratic harnesses,

We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a -commutation relation. This implies that quadratic harnesses are essentially determined uniquely by five numerical constants. Explicit recurrences for the orthogonal martingale polynomials are derived in several cases of interest.

q

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.

-commutations, and orthogonal martingale polynomials

Abstract Let X1,X2,… be a sequence of iid random variables having a continuous distribution; by R1,R2,… denote the corresponding record values. All the distributions allowing linearity of regressions either E(Rm+k|Rm) or E(Rm|Rm+k) are identified.

The bi-Poisson process: A quadratic harness

Asymptotic expansions of any order for expectations of inverses of random variables with positive binomial and negative binomial distributions are obtained in terms of the Eulerian polynomials. The paper extends and improves upon an expansion due to David and Johnson (1956-7).

Linearity of regression for non-adjacent record values

Behaviour of a sequence of independent identically distributed random variables with respect to a random threshold is investigated. Three statistics connected with exceeding the threshold are introduced, their exact and asymptotic distributions are derived. Also distribution-free properties, leading to some common and some new discrete distributions, are considered. Identification of equidistribution of observations and the threshold are discussed. In this context relations between the exponential and gamma distributions are studied and a new derivation of the celebrated Laplace expansion for the standard normal distribution function is given.

Asymptotic Eulerian expansions for binomial and negative binomial reciprocals

We use orthogonality measures of Askey-Wilson polynomials to construct Markov processes with linear regressions and quadratic conditional variances. Askey-Wilson polynomials are orthogonal martingale polynomials for these processes.

Distributional Properties of Exceedance Statistics

We find the distributions in R n for the independent random variables X and Y such that E(X|X + Y) = a(X + Y) and E(g(X)|X + Y) = bq(X + Y) where q runs through the set of all quadratic forms on R n orthogonal to a given quadratic form v. The essential part of this class is provided by distributions with Laplace transforms (1 - 2(c, s) + v(s)) -p that we describe completely, obtaining a generalization of a Gindikin theorem. This leads to the classification of natural exponential families with the variance function of type 1 pm ⊗ m-φ(m)M v , where M v is the symmetric matrix associated to the quadratic form v and m → φ(m) is a real function. These natural exponential families extend the classical Wishart distributions on Lorentz cones already considered by Jensen, and later on by Faraut and Koranyi.

Askey-Wilson polynomials, quadratic harnesses and martingales.

In this paper we study the property of linearity of backward regression for non-adjacent records. In the case of weak records, a characterization of the geometric distribution is obtained. It also appears that a related characterization for ordinary records does not hold, showing the difference in conditional behaviour between weak and ordinary records.

Laplace transforms which are negative powers of quadratic polynomials

Bayes negative binomial models under two different parameterizations are shown to be completely identifiable by the form of the Bayes estimates of the parameter. Also power series mixtures are briefly treated.