On integrability of quadratic harnesses

We investigate integrability properties of processes with linear regressions and quadratic conditional variances. We establish the right order of dependence of which moments are finite on the parameter defined below, raising the question of determining the optimal constant.


Introduction and main results
The study of processes with linear regression and quadratic conditional variances was initiated by Plucińska [Plucińska(1983)] who gave a characterization of the Wiener process in terms of conditional means and conditional variances. This approach was extended by several authors in many directions, sometimes with a trade-off between the amount of conditioning assumed and how much information is available about the covariance. Of the early generalizations, we mention here the work of Szab lowski [Szab lowski(1989)] who extracts information from minimal assumptions on conditioning, provided that the covariance is smooth. In his influential work Weso lowski [Weso lowski(1993)] characterized processes whose conditional variances is an arbitrary quadratic polynomial in the increments of the process; besides the Wiener process, such processes turned out to be either the Poisson process, or the negative binomial process, or the gamma process, or the hyperbolic secant process.
Processes with conditional variances given by more general quadratic polynomials have been analyzed in a series of papers that started with [Bryc(2001)], and culminated with Askey-Wilson polynomials in [Bryc and Weso lowski(2010)]. In these papers, the authors used algebraic techniques to study associated orthogonal polynomials, and to identify and construct a number of more exotic Markov processes with linear regressions and quadratic conditional variances.
The technique of orthogonal polynomials relies on apriori information about the existence of moments of all orders. Integrability can be deduced from assumptions on conditional variances, see [Bryc and Plucińska(1985)], [Weso lowski(1993)], and more recently [Bryc et al.(2007), Theorem 2.5]. Unfortunately, available integrability results do not cover all possible quadratic conditional variances, and in fact they do not use the full power of the "quadratic harness condition" (3.1) and (3.2) at all. Instead, they are based on the following weaker assumptions which appeared in [Bryc et al.(2007), (2.7),(2.8)(2.27), (2.28)]: For s < t, the one-sided conditional moments satisfy: and there are constants η, θ ∈ R and σ, τ ∈ [0, ∞) such that Our goal is to show that these expressions imply finiteness of moments of order 1/ √ στ , improving a bound in [Bryc et al.(2007), Theorem 2.5] who prove finiteness of moments of a logarithmic order − log(στ ).
Remark 1.3. Of course, if στ = 0 then moments of any order p > 0 are finite. Since this is already known from [Bryc et al.(2007), Theorem 2.5], here we concentrate on the case στ > 0.

Proof
The proof is based on tail estimates introduced in [Bryc and Plucińska(1985)]; the main improvement over [Bryc et al.(2007), Theorem 2.5] comes from a more careful choice of random variables X 1/u , X u with u close to 1. Such a choice was suggested by [Szab lowski(1986), Theorem 2].
We shall deduce Theorem 1.1 from the following result.
Proposition 2.1. Under the assumptions of Theorem 1.1, if 2 < p + 1 ≤ We first show how Theorem 1.1 follows, and then we will prove Proposition 2.1.
Proof of Theorem 1.1. It suffices to show that the moment of order p 0 = 1 240 √ στ exists. Without loss of generality we may assume that p 0 > 2, that is we consider only στ small enough. We apply Proposition 2.1 recursively.
2.1. Proof of Proposition 2.1. We first consider a pair of square-integrable random variables X, Y such that for some constants A, B ≥ 0. For small enough δ we have the following tail estimate.
Next, we prove the integrability lemma.

Integrability conjecture
Let δ > 0. Recall that (X t ) t∈T is a quadratic harness on a non-empty open interval T with parameters (η, θ, σ, τ, γ) if it satisfies (1.1), (1.2) for all s < t in T , and in addition that (1.3) and (1.4) hold with equality, and that for all s < t < u in T , Theorem 1.1 does not use (3.1), (3.2), so it does not use the all properties of a quadratic harness.
By Theorem 1.1, Conjecture 3.1 is true for στ = 0, and this result has essentially been known, although it was stated only for quadratic harnesses on (0, ∞). An even stronger version of Conjecture 3.1(ii), formulated by J. Weso lowski, says that if −1 ≤ γ < 1 − 2 √ στ and στ is small enough then X t is bounded. Here we indicate that this stronger version of Conjecture 3.1(ii) holds true for γ = −1.
Since γ = −1, the Hankel determinant is zero, and the distribution is concentrated on two points.