A survey on the effects of free and boolean convolutions on Cauchy-Stieltjes Kernel families

In the setting of noncommutative probability theory and in analogy with the theory of natural exponential families (NEFs), a theory of Cauchy-Stieltjes Kernel (CSK) families has been recently introduced. It is based on the Cauchy-Stieltjes kernel (1 − θx)−1. In this paper, after presenting some basic concepts on NEFs and CSK families and pointing out some similarities and differences between the two families, we review the present state and developments regarding the effects of free and boolean convolutions powers on CSK families. MSC2020 subject classifications: 60E10, 46L54.


Introduction
The theory of exponential families has received a great deal of attention in the classical probability and statistical literature and it remains a very interesting topic. This is in particular due to the fact that the most common distributions belong either to natural exponential families (NEFs) or to general exponential families. The notion of variance function is a fundamental concept in the theory of NEFs and many classifications of NEFs by the form of their variance function has been realized. The most important classes on R are the quadratic class of NEFs such that the variance function is a polynomial of degree less than or equal to two characterized by Morris [28] and the cubic class of NEFs such that the variance function is a polynomial of degree less than or equal to three characterized by Letac and Mora [25]. The multivariate version of the quadratic and cubic NEF's have been respectively described by Casalis [12] and Hassairi [23].
It is well known that the definition of a real NEF is based on the kernel (θ, x) −→ exp(θx). Wesolowski [36] has defined a notion of family generated by a measure ν for any kernel k(x, θ) such that converge in a open set Θ. It is the set of distributions {(k(x, θ)/L(θ))ν(dx) : θ ∈ Θ} .
Besides the exponential kernel, the most interesting example of kernels is the Cauchy-Stieltjes one (θ, x) −→ 1/(1 − θx). In fact, the authors in [9] have introduced the definition of q-exponential families, where they identified all the q-exponential families when |q| < 1. In particular, they studied the case where q = 0, which was related to the free probability theory by using the Cauchy-Stieltjes kernel 1/(1 − θx). When q = 1, we get the exponential families. Bryc [6] continued the study of Cauchy-Stieltjes Kernel (CSK) families for compactly supported probability measures ν. It was in particular shown that such families can be parameterized by the mean m. With this parametrization, denoting V (m) as the variance of the element with mean m, the function m → V (m) called the variance function and the mean m 0 of the generating measure ν uniquely determines the family and ν. The class of quadratic CSK families is described in [6]. This class consists of the free Meixner distributions. In [10], Bryc and Hassairi have extended the results established in [6] to allow probability measures with unbounded support. They have provided a method to determine the domain of means and introduced a notion of pseudo-variance function. They have also characterized a class of cubic CSK families with support bounded from one side. A general description of polynomial variance function with arbitrary degree is given in [8]. In particular, a complete description of the cubic compactly supported CSK families is given.
On the other hand, in the setting of non-commutative probability theory, Voiculescu introduce the notion of free independence. Moreover, if X and Y are free independent random variables with laws respectively denoted by μ and ν, then μ ν is the law of the sum of X and Y , where the operation is the free additive convolution which is defined using the R-transform. A multiplicative counterpart of free additive convolution, denoted by , was introduced in [5] for probability measures on the positive real line, and μ ν is the law of the product of X and Y . Speicher and Woroudi [32] have introduced a new kind of convolution between probability measures in the context of non-commutative probability theory with boolean independence: the boolean additive convolution . Moreover, if X and Y are boolean independent random variables with laws respectively denoted by μ and ν, then μ ν is the law of the sum of X and Y . A multiplicative counterpart of boolean additive convolution, denoted by ∪ × , was introduced by Bercovici [4], who showed how to calculate it using moment generating series.
In this paper we review some facts concerning the effects of free and boolean convolutions powers on CSK families. We present in section 2 some basic concepts about NEFs and CSK families. We provide some similarities and differences between the two families. In particular and in contrast to NEFs, a typical member of a given CSK family generates a different CSK family, so one can construct new CSK families by the iteration process. We relate the pseudo-variance function for the iterated family to the original pseudo-variance function, and we determine the domain of means. Section 3 is devoted to the study of free additive convolution from the perspective of CSK families. We present further similarities with NEFs and reproductive exponential models. We also explore a property of CSK families that have no counterpart in NEFs: We investigate when the domain of means can be extended beyond the natural domain. In section 4, we deal with boolean additive convolution from a point of view related to CSK families. We determine the formula for variance function under boolean additive convolution power. This formula is used to identify the relation between variance functions under boolean Bercovici-Pata bijection. We also give the connection between boolean cumulants and variance function and we relate boolean cumulants of some probability measures to Catalan numbers and Fuss Catalan numbers. In section 5, we focus on free multiplicative convolution. We determine the effect of the free multiplicative convolution on the pseudo-variance function of a CSK family. We then use the machinery of variance functions to establish some limit theorems related to this type of convolution and involving the free additive convolution and the boolean additive convolution. An explicit expression of the free multiplicative law of large numbers is also given. We are interested in section 6 on the boolean multiplicative convolution. We determine the effect of the boolean multiplicative convolution on the pseudo-variance function of a CSK family. We also identify the relation between variance functions under Belinschi-Nica type semigroup for multiplicative convolutions.

Cauchy-Stieltjes Kernel families
In the setting of non-commutative probability theory and in analogy with the theory of NEFs, a theory of CSK families has been recently introduced based on the Cauchy-Stieltjes kernel. In this paragraph we present some basic elements of CSK families. One start by presenting some basic concepts of NEFs. Then, we point out some similarities and differences between the two families.
To each μ in M(R) and θ in Θ(μ), we associate the following probability distribution: is called the natural exponential family (NEF) generated by μ. The measure μ is said to be a basis of F (μ). It is worth mentioning that a basis of F is by no means unique: If μ and μ are in M(R), then F (μ) = F (μ ) if and only if there exists (a, b) ∈ R 2 such that Therefore, all measure of the form (2.5) generate the family F , in particular the elements of F . In what follows, we will see that this property fails for CSK families. In fact a typical member in a CSK family generate something different than the original family, then the construction can be iterated. The map θ −→ κ μ (θ) is a bijection between Θ(μ) and its image M F which is called the domain of means of the family F. Denote by φ μ : M F −→ Θ(μ) the inverse of κ μ . We are thus led to the parametrization of F by the mean m.

About CSK families
Our notations are the ones used in [16]. Let ν be a non-degenerate probability measure with support bounded from above. Then is defined for all θ ∈ [0, θ + ) with 1/θ + = max{0, sup supp(ν)}. For θ ∈ [0, θ + ), we set is called the one-sided CSK family generated by ν. Let k ν (θ) = xP (θ,ν) (dx) denote the mean of P (θ,ν) . According to [10, page 579-580] the map θ → k ν (θ) is strictly increasing on (0, θ + ), it is given by the formula The image of (0, θ + ) by k ν is called the (one sided) domain of means of the family K + (ν), it is denoted (m 0 (ν), m + (ν)). This leads to a parametrization of the family K + (ν) by the mean. In fact, denoting by ψ ν the reciprocal of k ν , and writing for m ∈ (m 0 (ν), m + (ν)), Q (m,ν) (dx) = P (ψν (m),ν) (dx), we have that It is shown in [10] that the bounds m 0 (ν) and m + (ν) of the one-sided domain of means (m 0 (ν), m + (ν)) are given by , (2.10) where G ν (.) is the Cauchy transform of ν which is defined by It is worth mentioning here that one may define the one-sided CSK family for a measure ν with support bounded from below. This family is usually denoted K − (ν) and parameterized by θ such that If ν has compact support, the natural domain for the parameter θ of the two-sided CSK family K(ν) The variance function given by is a fundamental concept in the theory of CSK families as presented in [6]. Unfortunately, if ν hasn't a first moment which is for example the case for free 1/2-stable law, all the distributions in the CSK family generated by ν have infinite variance. This fact has led the authors in [10] to introduce a notion of pseudo-variance function defined by If m 0 (ν) = xdν is finite, then (see [10]) the pseudo-variance function is related to the variance function by (2.14) In particular, V ν = V ν when m 0 (ν) = 0. The generating measure ν is uniquely determined by the pseudo-variance function V ν . In fact, if we set (2.15) then the Cauchy transform (2.11) satisfies , (2.16) Also the distribution Q (m,ν) (dx) may be written as Now, we recall the effect on a CSK family of applying an affine transformation to the generating measure. Consider the affine transformation ϕ : x −→ (x − λ)/β where β = 0 and λ ∈ R and let ϕ(ν) be the image of ν by ϕ. In other words, if X is a random variable with law ν, then ϕ(ν) is the law of (X − λ)/β, or In particular, if β < 0 the support of the measure ϕ(ν) is bounded from below so that it generates the left-sided family K − (ϕ(ν)). For m close enough to (m 0 − λ)/β, the pseudo-variance function is In particular, if the variance function exists, then V ϕ(ν) (m) = 1 β 2 V ν (βm + λ). Note that using the special case where ϕ is the reflection ϕ(x) = −x, one can transform a right-sided CSK family to a left-sided family. If ν has support bounded from above and its right-sided CSK family K + (ν) has domain of means (m 0 , m + ) and pseudo-variance function V ν (m), then ϕ(ν) generates the leftsided CSK family K − (ϕ(ν)) with domain of means (−m + , −m 0 ) and pseudovariance function V ϕ(ν) (m) = V ν (−m). Remark 2.1. There are numerous similarities between the NEFs and the CSK families: both are parameterized by the mean, both are uniquely determined by the variance function and the so called "domain of means", and the variance function of the CSK family generated by the free additive convolution of generating measure ν has the same form as the variance function of the exponential family of the classical convolution (as we will see in the next section).
There also some differences due to the fact that the exponential kernel exp(θx) is always positive while the Cauchy kernel 1/(1 − θx) might be negative, and due to the fact that the variance of a CSK family might not exist. This fact has led the authors in [10] to introduce the "pseudo-variance" function that has no direct probabilistic interpretation but has similar properties to the variance function and is equal to the variance function of the CSK family generated by a measure ν of mean zero.

Iterated CSK families
One difference between the exponential and CSK families is that one can build nontrivial iterated CSK families. That is, each member of an exponential family generates the same exponential family so it does not matter which of them we use for the generating measure. But this is not so for CSK families: each member of a CSK family generates something different than the original family, so the construction can be iterated.

Theorem 2.3. [7, Theorem 2.3]
Let ν be a probability measure with support bounded from above, and let K + (ν) be the CSK family generated by ν. Fix m 1 ∈ (m 0 , m + ) and let B = B(ν) be given by (2.9). With the notations introduced above, we have (2.21) (ii) The (one sided) domain of means is .
Note that the function m −→ m is a bijection from D + (ν) into D + (Q (m1,ν) ), so that to get explicitly the pseudo-variance function of the CSK family K + (Q (m1,ν) ), we need to express m in terms of m from (2.21) and insert it in (2.22).
Note that as the probability measure Q (m1,ν) has a finite first moment m 0 = m 1 , the variance function V Q (m 1 ,ν) (.) of the CSK family K + (Q (m1,ν) ) exists and from (2.14) we have The following two special cases are of interest as they exhibit the iterated CSK families generated by two laws of importance in free probability. We consider the Wigner semicircle distribution. It is named after the Hungarian theoretical physicist Eugene Wigner who contributed to mathematical physics. The Wigner semicircle distribution arises as the limiting distribution of eigenvalues of a random symmetric matrices as the size of the matrix approaches infinity. In free probability theory, the role of Wigner's semicircle distribution is analogous to that of the Gaussian distribution in classical probability theory.
generates the CSK family with a constant variance function V ν (m) = 1 = V ν (m) and the (one-sided) domain of means is D + (ν) = (0, 1). (The full two-sided domain of means is of course (−1, 1).) For m 1 ∈ D + (ν), the probability measure generates the CSK family with pseudo-variance function and with the domain of means D + (Q (m1,ν) ) = (m 1 , 1 + m 1 ). The corresponding variance function is Up to an affine transformation, this is the Marchenko-Pastur law, see next example. In fact, in the mathematical theory of random matrices the Marchenko-Pastur distribution or Marchenko-Pastur law is introduced by the Ukrainian mathematicians Vladimir Alexandrovich Marchenko and Leonid Andreevich Pastur. It describes the asymptotic behavior of singular values of large rectangular random matrices, (see [26] for more details). Example 2.5. For 0 < a 2 < 1, the (absolutely continuous) centered Marchenko-Pastur (free Poisson) law generates the CSK family with quadratic variance function V (m) = 1 + am = V(m), and the domain of means is D + (ν) = (0, 1). For m 1 ∈ D + (ν), the probability measure generates CSK family with pseudo-variance function The domain of means is The variance function is

Free additive convolution
Free convolution is the free probability analog of the classical notion of convolution of probability measures. Due to the non-commutative nature of free probability theory, one has to talk separately about additive and multiplicative free convolution, which arise from addition and multiplication of free random variables. Free convolution can be used to compute the laws and spectra of sums or products of random variables which are free independents. Such examples include: random walk operators on free groups (Kesten measures) and asymptotic distribution of eigenvalues of sums or products of independent random matrices.
In this section, we are interested in the study of free additive convolution from the perspective of CSK families. Denote by M (respectively by M + ) the set of Borel probability measures on R (respectively on R + ). For ν ∈ M, its Cauchy transform G ν is defined by (2.11). Note in particular that (G ν (z)) < 0 for any z ∈ C + , and hence we may consider the reciprocal Cauchy transform F ν : C + −→ C + given by F ν (z) = 1/G ν (z) for z ∈ C + . According to [5], for any probability measure ν ∈ M and any λ ∈ (0, +∞), there exists positive numbers α, β and M such that F ν is univalent on the set Γ α,β := {z ∈ C + : (z) > β, | (z)| < α (z)} and such that F ν (Γ α,β ) ⊃ Γ λ,M . Therefore the right inverse F −1 ν of F ν exists on Γ λ,M , and the free cumulant transform R ν can be defined by The name refers to the fact that R ν linearizes free additive convolution (see [5]). Variants of R ν (with the same linearizing property) are the R-transform R ν and the Voiculescu transform v ν related by the following equalities: The free additive convolution μ ν of the probability measures μ, ν on Borel sets of the real line is a uniquely defined probability measure μ ν such that Our interest in the R-transform stems from its linear property to free additive convolution. Let ν α denote the α-fold free additive convolution of ν with itself. In contrast to classical convolution, this operation is well defined for all real α ≥ 1, (see [29]) and we have Probability measure ν is -infinitely divisible if its free additive convolution power ν α is well-defined for all real α > 0.

Free additive convolution and variance function
In this paragraph, we present further similarities of CSK families with exponential families and reproductive exponential models. Next, we give the formula of pseudo-variance function (and variance function in case of existence) by the effect of free additive convolution power. It is the same as the formula for variance function of a NEF under the effect of classical additive convolution power. More precisely: Proposition 3.10] Let V ν be the pseudo-variance function of the one sided CSK family generated by a probability measure ν with support bounded from above and with the mean −∞ ≤ m 0 < ∞. Then for α > 0 such that ν α is defined, the support of ν α is bounded from above and for m > αm 0 close enough to αm 0 , Furthermore, if m 0 < +∞, then the variance functions of the CSK families generated by ν and ν α exists and We remark that the restriction of (3.5) to m close enough to αm 0 cannot be easily avoided, as we do not have a general formula for the upper end of the domain of means for ν α . The study of the action of free additive convolution power on the domain of means is given in what follows, (see [7] for more details).
A property which in [3, (3.16)] is indeed called the reproductive property of an exponential family states that if μ ∈ F with variance function V F , then for all n ∈ N the law of the sample mean, D 1/n (μ * n ) is in the NEF with variance function V F /n. The analogue of this result for the CSK families is given in [6], [10]. More precisely we have, Proposition 3.2. Suppose V ν is the pseudo-variance function of the CSK family K + (ν) generated by a non degenerate probability measure ν with support bounded from above and mean m 0 (ν), then for α ≥ 1 measure has also support bounded from above and there is ε > 0 such that the pseudovariance function of the one sided CSK family generated by ν α is If ν is -infinitely divisible, then the above holds for every Recall that if ν is a compactly supported measure, the R-transform is analytic The coefficients c n = c n (ν) are called free cumulants of measure ν. The following result give the link between free cumulants and variance function of a CSK family, (see [6] for more details).
We now use (3.9) to relate certain free cumulants to Catalan numbers, see

Marchenko-Pastur approximation
Let denote the semicircle law of mean a and variance σ 2 . Up to affine transformations, this is the free Meixner law which generates the CSK family with the variance function V ωa,σ (m) = σ 2 . Following the analogy with NEFs, the CSK family K(ω a,σ ) can be thought as a free analog of the NEF generated by the normal distribution. Somewhat surprisingly, this family does not contain all semicircle laws, but instead it contains affine transformations of the (absolutely continuous) Marchenko-Pastur laws.
Function V (m) = 1/λ is the variance function of the CSK family To verify that the expression integrates to 1 for m = 0, we use the explicit form of the density [ By Example 3.5, if 0 < |m| ≤ σ, then up to affine transformation π m,1/σ 2 is a Marchenko-Pastur law. Thus in this case Theorem 3.6 gives a Marchenko-Pastur approximation to L(Y λ ).
Of course, every compactly supported mean-zero measure ν is an element of the CSK family that it generates. Since π 0,1/σ 2 = ω 0,σ is the semicircle law, combining Proposition 3.2 with Theorem 3.6 we get the following Free Central Limit Theorem; see [11], [35]. Corollary 3.7. If a probability measure ν is compactly supported and centered, then with σ 2 = x 2 ν(dx), we have

Extending the domain for parametrization by the mean
We investigate when the domain of means can be extended beyond the natural domain. This is a property in CSK families that have no counterpart in NEFs. Given a probability measure ν with support bounded from above, equation (2.10) tells us how to determine the one-sided domain of means (m 0 , m + ) and formula (2.13) tell us how to compute the pseudo-variance function V ν (m) for m ∈ (m 0 , m + ). But the pseudo-variance function is often well defined for other values of m, too. So it is natural to ask whether the corresponding "family of measures" can also be enlarged preserving the pseudo-variance function.
The following examples illustrates the idea.
Example 3.8. Consider the (two-sided) CSK family generated by the semicircle , the domain of means (−1, 1) and This is a family of atomless Marchenko-Pastur laws, which can be naturally enlarged to include all Marchenko-Pastur laws: Noting that we see that V ν (m) = 1 is the variance function of this enlarged family.
Of course, it may also happen that the extension beyond the natural domain of means is not possible. Here is a simple example when this happens. 1). So the domain of means here is (−1, 1), and with m 0 (ν) = 0 the pseudo-variance function is equal to the variance function, In this case, the variance function is negative outside the domain of means, so we cannot extend the family {Q (m,ν) : m ∈ (−1, 1)} beyond the original domain of means.
Our next example shows that the extension sometimes may proceed in two separate steps. Example 3.10. Consider the inverse semicircle law Since is non-negative and well defined for all m. Since the integral Q (m,ν) (dx) is an analytic function of m < −1/2, it must be 1, so Q (m,ν) is a probability measure for all m < −1/2. This is the "first part" of the extension, from (−∞, −1) to a larger interval Inspecting the original definition of P (θ,ν) we see that the kernel 1/(1 − θx) is positive on the support of ν for θ from Θ = (0, ∞), which was the set used in the definition, but it is also well defined for θ < −1/4. So this extension "includes" this second set, with m = −1 corresponding to infinite values of θ.
At m = −1/2 the integrand has singularity at x = −1/4 but the integral is still 1, see the calculation below. For m > −1/2, the mass becomes less then one, with the extra mass in the atomic part, which located at m + m 2 so that the mean is preserved.
We now prove the above two claims in Example 3.10.

Proof. By the change of variable
we obtain dt.
The integrand can be decomposed as follows .
We now verify that the atomic part works as needed. By the change of variable We now give a general theory that shows how the two-step extension works.

The first extension
Suppose that the pseudo-variance function V ν (.) extends as a real analytic function to (m 0 , +∞). Denote by A = A(ν) = sup supp(ν), recall notation (2.9) and define We know that m + (ν) ≥ m + is well defined. We will verify that one can use  The rest of this paragraph contains proof of Theorem 3.11. We consider the set Θ for which the transform (2.6) exists. In fact, if A(ν) ≥ 0, then Θ = (0, θ + ) with θ + = 1 B , and if A(ν) < 0, then One can always write One can then define the first extension of K + (ν) as Note that K + (ν) = K + (ν) when A(ν) ≥ 0, because in this case Therefore, the first extension is non-trivial only when A(ν) < 0.
We then define the function ψ ν on (m 0 , m + ) as the inverse of the restriction of k ν (.) to (0, ∞), and on (m + , m + (ν)) as the inverse of the restriction of k ν (.) to −∞, 1 A(ν) . This leads to the parametrization by the mean m ∈ (m 0 , m + ) ∪ (m + , m + (ν)) of the family K + (ν). The definition of the pseudo-variance function can also be extended using the function ψ ν . Following (2.13), we define V ν (.) for m ∈ (m 0 , m + ) ∪ (m + , m + (ν)) as We have that , The explicit parametrization by the means of the enlarged family can then be given by

Domain of means under affine transformation
Let ϕ an affine transformation. It is well known that the lower end of the one sided domain of means of the family K + (ν) behave nicely under the action of affine transformation ϕ, that is m 0 (ϕ(ν)) = ϕ(m 0 (ν)). But we do not have a general formula of the upper end for the natural domain of means for K + (ν).
The following examples show that there is no simple formula for m + (ν) under affine transformation.  1 4 ) (x)dx. (3.17) and it generates the CSK family with pseudo-variance function V ϕ(ν) (m) = m(m − 1/2) 2 . We have that
Example 3.14. Consider the (two-sided) CSK family generated by the semicircle law with the variance function V ν (m) = V ν (m) = 1 and domain of means (−1, 1), that is With B(ϕ(ν)) = max{0, sup supp(ϕ(ν))} = 0, the (two-sided) range of parameter is (θ − , θ + ) = (−1/5, +∞). The probability measure ϕ(ν) generates the (two-sided) CSK family We have The purpose is to give a more natural definition for the domain of means of a CSK family that behave nicely under affine transformation. In several references, we consider the range of the parameter θ such that 1/θ ∈ (sup supp ν, ∞) ∩ [0, ∞). In fact authors in [10] have pushed forward the theory of CSK families by extending the results in [6] to allow measures ν with unbounded support. In such situation, the family is parameterized by a 'one-sided' range of θ of a fixed sign, so that generating measures have support bounded from above and the CSK families are parameterized by θ > 0, which gives the domain of means (m 0 , m + ). We can include additional range of θ which is possible only when the support of ν is in (−∞, 0). In this case we can include additional range of 1/θ ∈ (sup supp ν, 0), so the extended range of θ would have a simpler description Θ(ν) = {θ; 1/θ ∈ (sup supp ν, ∞)}, that is, Θ(ν) is the set for which the transform M ν exists and, with A = A(ν) = sup supp(ν), it can be written as This extension for the range of the parameter θ was considered in the first extension.

The second extension
As indicated by Example 3.14 and Example 3. It is clear that M + ≥ m + . In fact, M + ≥ m + . This can be seen from (3.12): since the mean must be smaller than A(ν) we have m + ≤ A(ν), so V ν (m)/m ≥ 0 for all m < m + . It is easy to see that M + = ∞ > m + in Example 3.14 and in Example 3.10 while M + = m + = m + in Example 3.9. We now introduce the second extension of K + (ν) as the family of measures with Q (m,ν) given by where the weight of the atom is It is clear that the expression on the right hand side of (3.23) is well defined at all m such that m + V ν (m)/m > A. We need to show that the expression is well defined also at the points where m + V ν (m)/m = A; one such point is of course m + . The argument here relies on the fact that G ν is analytic in the slit plane C\ (−∞, A). Furthermore, G ν (a) is decreasing to 0 and convex on (A, ∞). In particular lim a−→A + G ν (a) exists, and is either ∞ or m + /V ν (m + ). Furthermore, if the limit is ∞, then V(m + )/m + = 0, which implies that M + = m + . So without loss of generality we may assume that lim a−→A + G ν (a) = m + /V ν (m + ) < ∞. and that the integral defining G ν (A) converges.
Suppose m 1 < M + such that m 1 + V ν (m 1 )/m 1 = A. Then V ν (m 1 )/m 1 = A − m 1 > 0 and, taking the limit, On the other hand, Therefore, for m ∈ [m + , M + ) such that m + V ν (m)/m = A, the right hand side of (3.23) is well defined and simplifies to Then, the second extension of the family is given by Since formula (2.16) holds for all m ∈ (m 0 , m + ), it is clear that K + (ν) ⊂ K + (ν). We now verify that the extension satisfies desired conditions. is Here the use of V ν (m) is based on the assumption the pseudo-variance function V ν extends as a real analytic function to (m 0 , +∞).

Domain of means under free additive convolution power
One notes that the lower end of the one sided domain of means of the family K + (ν) satisfies the relation, for α > 0 We have that V ν (m) = 1 and m + (ν) = 1. Then μ = ν ν is the semicircle law with density The CSK family generated by μ has a pseudo-variance function V μ (m) = 2.
It is well known that the domain of means for exponential families scales nicely under classical additive convolution power, and it is satisfying to note that the domain of means of the extended CSK family K + (ν) lead to the analogous formula: Indeed, since V ν α (m) = αV ν (m/α), the result follows from (3.22).

Boolean additive convolution
Let ν ∈ M. The boolean additive convolution is determined by the K-transform K ν of ν which is given by The function K ν is usually called self energy and it represent the analytic backbone of boolean additive convolution. For two probability measures μ and ν in M, the boolean additive convolution μ ν is determined by and μ ν is again a probability measure. According to [32], we call a probability measure ν ∈ M is infinitely divisible in the boolean sense, if for each n ∈ R, there exists ν n ∈ M such that Note that all probability measure ν ∈ M are -infinitely divisible, see [32,Theorem 3.6].

Boolean additive convolution and variance function
In this paragraph, we deal with boolean additive convolution from the perspective of CSK families. Next, we give the formula for pseudo-variance function (and variance function V ν in case of existence) under boolean additive convolution power.
Theorem 4.1. [16,Theorem 2.3] Suppose V ν is the pseudo-variance function of the CSK family K + (ν) generated by a non degenerate probability measure ν with support bounded from above and mean m 0 (ν). For α > 0, we have that: (i) The support of ν α is bounded from above.
(ii) For m close enough to αm 0 (ν), Furthermore, if m 0 < +∞, then the variance functions of the CSK families generated by ν and ν α exists and
The following result gives formulas for pseudo-variance functions (and variance functions in case of existence) under both free additive convolution and boolean additive convolution power.

Proposition 4.3. [16, Proposition 2.6] Suppose
V ν is the pseudo-variance function of the CSK family K + (ν) generated by a non degenerate probability measure ν with support bounded from above. For α > 0 such that probability measures ν 1/α α and ν 1/α α are well defined, their support are bounded from above and they generates CSK families with pseudo-variance functions

5)
and respectively, for m close enough to m 0 . Furthermore, if m 0 < +∞, then the variance functions of the CSK families generated respectively by ν, ν 1/α α and ν 1/α α exists and for m close enough to m 0 we have and Authors in [2] consider the transformation B t : M −→ M defined by, for every t ≥ 0  Denote by V the class of variance functions corresponding to probability measures ν such that ν is compactly supported, centered: xν(dx) = 0, with variance x 2 ν(dx) = 1, so that V ν (0) = 1. Denote V ∞ the class of those V ∈ V that the function m → V (cm) is in V for every real c. It was proved in [8, We will see that this bijection between variance functions correspond to the boolean Bercovici-Pata bijection between probability measures.

to the inverse boolean Bercovici-Pata bijection between probability measures
If ν is a compactly supported probability measure on the real line, the Ktransform K ν of ν admit a Laurent expansion. From [32], one sees that The coefficients r n = r n (ν) are called the boolean cumulants of the measure ν. In particular r 0 = 0, r 1 = xν(dx) = m 0 . The following result gives the connection between boolean cumulants and variance functions of CSK families. (4.16) In the following we relate boolean cumulants of the Marchenko Pastur distribution to Catalan numbers. The centered Marchenko-Pastur distribution is given by The discrete part is absent except for a 2 > 1, in this case p 1 = 1 − 1/a 2 and x 1 = −1/a. It generates the CSK family with variance function V ν (m) = 1 + am = V ν (m).

Corollary 4.7. [16, Corollary 3.2] If ν is the centered standardized Marchenko
Pastur distribution with parameter a = 2, i.e. it generates the CSK family with m 0 = 0 and variance function V ν (m) = 1 + 2m, then its boolean cumulants are r 0 = 0, r 1 = m 0 = 0 and for n ≥ 1, Next, we relate boolean cumulants of certain probability distribution to Fuss-Catalan numbers. In combinatorial mathematics and statistics, the Fuss-Catalan numbers are defined in [21] by the Swiss mathematician Fuss, Nicolaus. They are numbers of the form The Fuss-Catalan represents the number of legal permutations or allowed ways of arranging a number of articles, that is restricted in some way. This means that they are related to the Binomial coefficient.
On the other hand, some examples of variance functions that are polynomial in the mean of arbitrary degree are introduced in [8]. In particular a complete resolution of compactly supported CSK with cubic variance function is given (see [8,Theorem 1.2]). Next, we relate boolean cumulants of certain probability distribution generating a cubic CSK family to Fuss-Catalan numbers of the form (4.18) for p = 3 and r = 1.

Some approximations in CSK family
In this paragraph, we give an approximation of elements of the CSK family generated by the boolean Gaussian distribution and an approximation of elements of the CSK family generated by the boolean Poisson distribution, (see [20] for more details).

Approximation of boolean Poisson CSK family
For N ∈ N, s > 0 and 0 < λ < N, consider

R. Fakhfakh
We have that for all θ ∈ (−∞, 1 s ), As the inverse of the function k μ N (.), we have that for all m ∈ (0, s) = k μ N ((−∞, Formula (2.13) implies that the pseudo-variance function of the two sided CSK family K(μ N ) is With m 0 (μ N ) = λs/N , we see from (2.14) that the variance function of the two sided CSK family K(μ N ) is The CSK family generated by μ N is given by The boolean Poisson distribution π (s) λ with jump size s and parameter λ (s, λ ≥ 0) is given by We have for all θ ∈ (−∞, 1 s(λ+1) ) and k π (s) λ (θ) = λs 1 − θs .
The following result gives an approximation of elements of the boolean Poisson CSK family. In particular we get the boolean Poisson limit theorem, (see [32,Theorem 3.5]).
Theorem 4.11. [20] For N ∈ N, s > 0 and 0 < λ < N, let and consider the CSK family generated by μ N N , with mean m 0 (μ N N ) = λs and variance function V μ N N (.). We have that , in distribution. for all m in a neighborhood of m 0 = λs. In particular, for m = m 0 = λs, we get the boolean Poisson limit theorem

Free multiplicative convolution and variance function
In this paragraph, we deal with free multiplicative convolution from a point of view related to CSK families. We first state the result concerning the effect of the free multiplicative convolution power on a CSK family.
Theorem 5.1. [22] Let V ν be the pseudo-variance function of the CSK family K − (ν) generated by a non degenerate probability distribution ν concentrated on the positive real line with mean m 0 (ν). Consider α > 0 such that ν α is defined. Then (ii) If m 0 < +∞, then the variance functions of the CSK families generated by ν and ν α exist and Several limit theorems involving the free additive convolution, the boolean additive convolution and the free multiplicative convolution have been established in [27] and in [30]. The authors in [22] used variance functions to re-derive these results, this leads to some new variance functions with non usual form.
Proposition 5.4. [22] Let ν be a non degenerate probability distribution concentrated on the positive real line with mean m 0 (ν) > 0. Suppose that ν has a finite second moment. Then denoting γ = V ar(ν) (m0(ν)) 2 = Vν (m0) It is worth mentioning that the R and R-transforms of the limiting distribution η γ , is given in [30], in terms of the Lambert's W -function which satisfies the functional equation z = W (z) exp(W (z)).
For more details of the Lambert W -function, see [13]. Let W 0 (z) be the principal branch of the Lambert W -function.
Proposition 5.6. The probability density ω σ of the measure σ can be given in the implicit form as: , 0 < v < π.

Explicit free multiplicative law of large numbers
The limit probability measure for the free multiplicative law of large numbers was proved by Tucci [33] for probability measures with bounded support. Haagerup for all t ∈ (α, 1) and μ({0}) = α. The support of the measure μ is the closure of the interval Consider V ν the pseudo-variance function of the CSK family K − (ν) generated by a non degenerate probability measure ν ∈ M + . We give explicitly the law of large numbers μ for free multiplicative convolution in terms of the pseudovariance function V ν .

Boolean multiplicative convolution
For ν ∈ M + , the η-transform of ν is defined by: .