ELECTRONIC COMMUNICATIONS in PROBABILITY

We study the Laha-Lukacs property of the free Meixner laws (processes). We prove that some families of free Meixner distribution have the linear regression function. We also show that this families have the property of quadratic conditional variances.


Introduction
The original motivation for this paper comes from a desire to understand the results on conditional expectation in the work of Bożejko and Bryc [7].They proved, that if the first conditional moment is a linear regression and conditional variances are quadratic functions, then the corresponding variables have free Meixner type laws (theorem 3. 2).An open problem in this area is a converse of their theorem.We will show that free Meixner variables satisfy the condition from theorem 3.2 of [7].In particular, we will apply this result to describe characterization of free Lévy processes.It is natural to study relations between classical and free probability.We will present a theorem which is the free non-commutative analog of the classical result by Wesolowski [15,16].Let us mention that he followed the argument in [9].Laha and Lukacs in [9] characterized all the (classical) Meixner distributions using a quadratic regression property.Wesołowski proved that in classical probability the quadratic conditional variance characterize a subclass of Lévy processes.Similar results have been obtained in boolean probability by Anshelevich [2].He showed that in the boolean theory the Laha-Lukacs property characterizes only the Bernoulli distributions.It is worthwhile to mention the work of Bryc [8], where the Laha-Lukacs property for q-Gaussian processes was shown.Bryc proved that classical processes corresponding to operators which satisfy a q-commutation relations have linear regressions and quadratic conditional variances.For q = 0 we have the free case, so that his result is a special case of the free Wigner's semicircle elements, which we consider.

Free Meixner laws, free cumulants and conditional expectation
Classical Meixner distributions first appeared in the theory of orthogonal polynomials in the work of Meixner [10].In free probability the Meixner systems of polynomials were introduced by Anshelevich [1], Bożejko, Leinert, Speicher [6] and Saitoh and Yoshida [12].They showed that free Meixner system can be classified into six types of laws: the Wigner semicircle, the free Poisson, the free Pascal (free negative binomial), the free Gamma, a law that we will call pure free Meixner and the free binomial law.
We assume that our probability space is a von Neumann algebra A with a normal faithful tracial state τ : A → C i.e., τ () is linear, weak*-continuous and τ (XY) = τ (YX), τ (I) = 1, τ (XX * ) ≥ 0 and τ (XX * ) = 0 implies X = 0 for all X, Y ∈ A. A (noncommutative) random variable X is a self-adjoint (X = X * ) element of A. We are interested in the two-parameter family of compactly supported probability measures (so that their moments grow at a geometric rate) {µ a,b : a ∈ R, b ≥ −1} with the Cauchy-Stieltjes transform given by the formula where the branch of the analytic square root should be determined by the condition that Im(z) > 0 ⇒ Im(G µ (z)) 0 (see [12]).Cauchy-Stieltjes transform of µ is a function G µ defined on the upper half plane C + = {s + ti|s, t ∈ R, t > 0} and takes values in the lower half plane C − = {s + ti|s, t ∈ R, t ≤ 0}.
Equation (2.1) describes the distribution with mean zero and variance one (see [12]).The moment generating function, which corresponds to the equation (2.1), has the form for |z| small enough.The R-transform of a random variable , where R i (X) is a sequences defined by (2.4) (see [4] for more details).For reader's convenience we recall that the R-transform corresponding to M (z) is equal to where the analytic square root is chosen so that lim z→0 R µ (z) = 0 (see [12]).If X has the distribution µ a,b , then sometimes we will write R X for the R-transform of X .For particular values of a, b the law of X is: • the Wigner's semicircle law if a = b = 0; • the free Poisson law if b = 0 and a = 0; • the free Pascal (negative binomial) type law if b > 0 and a 2 > 4b; • the free Gamma law if b > 0 and a 2 = 4b; • the pure free Meixner law if b > 0 and a 2 < 4b; • the free binomial law −1 ≤ b < 0.
Random variables X 1 , . . ., X n are freely independent (free) if, for every n ≥ 1 and every non-constant choice of Y i ∈ {X 1 , . . ., X n }, where i ∈ {1, . . ., k} (for some positive integer k) The following theorem shows connection between the Cauchy transform and the Rtransform.Part (B) describes additive free convolution.We shall apply this theorem without further comment (it can be found in Nica, Speicher [11]). (2.6) The R-transform linearizes the free convolution, i.e. if µ and ν are (compactly supported) probability measures on R, then we have where denotes the free convolution (the free convolution of measures µ, ν is the law of X + Y where X, Y are free and have laws µ, ν respectively).
Below we introduce Lemma 4.1 of [7], which we will use in the main theorem to calculate the moment generating function of free convolution.
If B ⊂ A is a von Neumann subalgebra and A has a trace τ , then there exists a unique conditional expectation from A to B with respect to τ , which we denote by τ (.|B).This map is a weakly continuous, completely positive, identity preserving, contraction and it is characterized by the property that, for any X ∈ A, τ (XY ) = τ (τ (X|B)Y ) for any Y ∈ B (see [5,14]).For fixed X ∈ A by τ (.|X) we denote the conditional expectation corresponding to the von Neumann algebra B generated by X.The conditional variance is defined as usual V ar(X|B) = τ ((X − τ (X|B)) 2 |B). (2.9) The following lemma has been proven in [7].

The main result
The following is our main results of the paper.
Proof.First we compute the law of X + Y.
Similarly we compute R-transform of the variable Y.This allows us to find the Rtransform of X + Y (assuming b = 0) From equations (3.3) and (3.4) it follows that Analogously we get (3.9) The fact that X and Y are freely independent implies that We can thus compute the power series

Theorem 2 . 1 .
(A)The relation between the Cauchy transform G µ (z) and the R µ (z)transform of a probability measure µ is given by

Lemma 2 . 3 .
Let W be a (self-adjoint) element of the von Neumann algebra A, generated by a self-adjoint V ∈ A. If, for all n ≥ 1 we have