ELECTRONIC COMMUNICATIONS in PROBABILITY

In this article we give a purely noncommutative criterion for the characterization of free Meixner random variables. We prove that some families of free Meixner distributions can be described in terms of the conditional expectation, which has no classical analogue. We also show a generalization of Speicher's formula (relating moments and free cumulants) and establish a new relation in free probability.


Introduction
Classical Meixner distributions appeared in the work of Meixner [22] on the theory of orthogonal polynomials.In free probability the Meixner systems of polynomials were introduced by Anshelevich [2] and Saitoh and Yoshida [24].The study of free Meixner distributions has been an active field of research during the last decade -see works [3,4,9,10,11,14,20].It is common in free probability, that their properties, to a large extent, are analogous to those of the classical Meixner distributions.This especially regards their characterizations: in both cases it is achieved in terms of generating functions of the polynomials and the quadratic regression property.The main aim of this paper is to produce a new characterization of the free Meixner laws which is close to the quadratic regression property, but with no analog to classical Meixner distributions.The quadratic regression property for free Meixner distributions has been established by Bożejko and Bryc [9] -see also Ejsmont [17] for the reverse part.Shortly this says that the first conditional moment is a linear regression and conditional variance is quadratic if and only if corresponding variables have free Meixner distributions.
As an example we can give two random variables which have the same distribution (because the result is more transparent with this assumption).Suppose that X,Y are free, self-adjoint, non-degenerate, centered and have the same distribution.Then the X and Y have free Meixner laws (with respect to some state τ , which we will discuss later in section 2) if and only if there exist constant a, b such that Loosely speaking, the formula (1.1) says that we can discard (X + Y) linearly from the right side of equation (1.1) in the front of the conditional expectation if and only if X and Y have free Meixner distribution.This result is unexpected because in commutative probability equation (1.1) takes the form for any classical variables X and Y.In particular, we apply fact (1.1) to prove the main result of this paper about characterization of free Lévy processes.It is worth mentioning here that a Laha-Lukacs type characterizations of random variables in free probability are also studied by Szpojankowski, Wesołowski [27] and Bryc [13].The first authors give a characterization of noncommutative free-Poisson and free-Binomial variables by properties of the first two conditional moments, which mimics Lukacs type assumptions known from classical probability.Bryc in [13] proved that q-Gaussian processes have linear regressions and quadratic conditional variances.The paper is organized as follows.In section 2 we review basic free probability, free Meixner laws and the statement of the main result.In the third section we look more closely at non-crossing partitions with first two elements in the same block.In this section we also give a new characterization of free Meixner systems and we generalize the Speicher's identity.Finally, in Section 4 we prove our main results.

Free probability and Meixner laws
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, continuous in weak* topology, τ (XY) = τ (YX), τ (I) = 1, τ (XX * ) ≥ 0 and τ (XX * ) = 0 implies X = 0 for all X, Y ∈ A. A (noncommutative) bounded random variable X is a self-adjoint (i.e.X = X * ) element of A. We are interested in the two-parameter family of compactly supported probability measures (so that their moments do not grow faster than exponentially) {µ a,b : a ∈ R, b ≥ −1} with the moment generating function given by the formula for |z| small enough (the branch of the analytic square root should be determined by the condition that (z) > 0 ⇒ (G µ (z)) 0 (see [24]).Equation (2.1) describes the distribution with mean zero and variance one (see [24]).For particular values a, b we have six types of distribution: 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 (see [9,17] for more details).
Let C X 1 , . . ., X n denote the non-commutative ring of polynomials in variables X 1 , . . ., X n .The free (non-crossing) cumulants are the k-linear maps R k : C X 1 , . . ., X k → C defined by the recursive formula (connecting them with mixed moments) where and N C(n) is the set of all non-crossing partitions of {1, 2, . . ., n} (see [23,26]).Sometimes we will write R k 2) (see [7,23] for more details).If X has the distribution µ, then sometimes we will write R µ for the R-transform of X.
Random variables X 1 , . . ., X n are freely independent (free) if, for every k ≥ 2 and every non-constant choice of Y i ∈ {X 1 , . . ., X n }, where i ∈ {1, . . ., k} (for some positive integer k) 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).
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 and Y ∈ B, τ (XY) = τ (τ (X|B)Y ) (see [8,28]).For a 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). (2.5) A non-commutative stochastic process (X t ) is a free Lévy process if it has free additive and stationary increments.For a more detailed discussion of free and classical Lévy processes with finite moments of all orders we refer to [6,21].Let us first recall some properties of free Lévy processes which follow from [9].If (X t ) is a free Lévy process such as τ (X t ) = 0 and τ (X for all 0 < s < u.The conditional variance for free Lévy process is equal (for 0 < s < u) (2.7) For more details about free convolutions and free probability theory, the reader can consult [19,23,29].

The main result
The main result of this paper is the following characterization of free Meixner processes in terms of the conditional expectation.The proof of this theorem is given in Section 4.
Theorem 2.1.Suppose that (X t≥0 ) is a free Lévy process such that τ (X t ) = 0 and τ (X 2 t ) = t for all t > 0. Then the increment (X t+s − X t )/ √ s (t, s > 0) has the free Meixner law µ a/ √ s,b/s (for some b 0) if and only if for all t < s τ (2.8) Remark 2.2.The existence of a free Lévy process was demonstrated by Biane [8] who proved that from every infinitely divisible distribution we can construct a free Lévy process.We assume that b 0 in Theorem 2.1 because a free Meixner variable is infinitely divisible if and only if b 0 (see [5,9]).
The proof of Theorem 2.1 is based on the following fact.
• the normalized Kesten law if and only if (b = 0 and a = 0)

Complementary facts and indications
We need the following lemmas on conditional expectations to prove the main result.The lemmas 2.5 and 2.6 were proved in [9] and [18], respectively.
Lemma 2.6.If X and Y are free independent and centered, then the condition βR k (X) = αR k (Y) for β, α > 0 and all integers k is equivalent to and the following functions (power series): for sufficiently small |z| < and z ∈ C.
Remark 2.8.This series is convergent because we consider compactly supported probability measures, so moments and cumulants do not grow faster than exponentially (see [7]).This implies that c (k) n also does not grow faster than exponentially.Now we introduce a lemma which we will use in the proof of the theorems 3.2, 3.6 and 2.3 (for the proof see [18]).
Lemma 2.9.Let Z be a (self-adjoint) element of the von Neumann algebra A then (2.13) Below we recall some results of [9], which we will apply in the proof of the main theorem to calculate the moment generating function of free convolution.

A generalization of Speicher's identity
By the main result of [25], we have the following relation The relation (3.1) can be generalized as following: Proposition 3.1.Suppose that X is a self-adjoint element of the algebra A and denote by µ the distribution of X, then where R (k) Proof.We prove this by the induction on k.The case k = 1 is clear because C (1) follows immediately by using Lemma 2.9 which gives (zM (z))z k+1 M (z).

A new relation between moments of free Meixner laws
Any probability measure µ on the real line, whose all moments are finite, has two associated sequences of Jacobi parameters α i , β i for example, µ is the spectral measure of the tridiagonal matrix (see [5,16]) We will denote this fact by If the measure µ has all finite moments, then by a theorem of Stieltjes (see [1]), its Cauchy-Stieltjes transform can be expressed as a continued fraction: If some β i = 0, the continued fraction terminates, in which case the subsequent α and β coefficients can be defined arbitrarily.See [12,16] for more details.The monic orthogonal polynomials P n for µ satisfy a recursion relation with −1 (x) = 0.
Thus it is natural to ask about the relation between measures whose Jacobi parameters are described by (3.4) and some other measures whose Jacobi parameter are equal Theorem 3.2.Suppose that X is a self-adjoint element of the algebra A and denote by µ the distribution of X.If the measure µ has Jacobi parameters described by (3.4) where β 1 > 0, then the relation between the measure ρ of the variable Y described by parameter (3.7) is given by for all n 0 .
Proof.From (3.5) we have Using the relations Applying Lemma 2.9 to k = 1 we get where C µ (z) is function for X.Now we apply (3.11) to the equation (3.10) and after a simple computation, we obtain which is equivalent to (3.8) and this completes the proof.
n (µ) is the moment of the variable described by Jacobi parameters (3.7).
Remark 3.4.The normalized free Meixner distributions µ a,b have Jacobi parameter in other words their Jacobi parameters are independent of n for n ≥ 1 (see also [5]).Proof.Under the assumption that X is a free Meixner random variable we have that Jacobi parameter given in (3.13).By Theorem 3.2 we get that c (see also [5,9]).
⇐: Let's suppose, that equality (3.16) holds.Multiplying (3.16) by z n+2 for n 0 we obtain (m 1 = 0 and m 2 = 1) where M (z) is the moment generation function for the Wigner semicircle law with mean a and variance b + 1.The above equation is equivalent to It is well known (see [9,23]) that  [17].From this theorem we also have αR k (Y) = βR k (X).From Lemma 2.10 the moment generating functions M of X + Y satisfy equation (2.14).If in (2.14) we multiply the both sides by (1 − C (2) (z)) and use Lemma 2.9 with k = 1 where C (2) (z) is function for X + Y. Now we apply Lemma 2.9 with k = 2 to equation (4.1) (using the assumption R 2 (X + Y) = 1) and after simple computations, we see that    Open problems and remarks.In Theorem 2.1 of this paper we assume that random variables are bounded that is X t ∈ A. It would be interesting to show if this assumption can be replaced by X t ∈ L 2 (A, τ ).It would be also worth to investigate if Theorem 3.6 is related to the main result of [15].

Proposition 3 . 5 .
Suppose that X is a self-adjoint element of the algebra A and have the free normalized Meixner law µ a,b where b > −1.Then c(2)  n (µ a,b ) = x n dw a,b+1 (dx),(3.14)for all n 0 and w a,b is the Wigner semicircle law with mean a and variance b.