Some Limiting Laws in Non-Commutative Probability

: In this article, we provide some new limiting laws related to the free multiplicative law of large numbers and involving free and Boolean additive convolutions. Some examples of these limiting laws are presented within the framework of non-commutative probability theory


Introduction
In recent decades, a number of articles have studied limit theorems with respect to the free convolution of probability measures.The concept of freeness is the key notion.It may be viewed (for non-commutative random variables) as a type of independence.Like in classical probability, in which the notion of independence leads to classical convolution, the notion of freeness gives rise to a binary operation for real measures: the free convolution.A lot of classic results for the addition and multiplication of independent random variables possess analog properties for this new theory, for example, the central limit theorem, the Lévy-Khintchine formula, the law of large numbers and others.For an introduction to these subjects, we refer to [1].In [2], the authors provide the distributional model behavior of the sum of free random variables distributed in an identical manner.They explicitly describe the relation between the limiting laws for classical and free additive convolutions.On the other hand, for measures with bounded support, Tucci [3] proved the limiting distribution for the free multiplicative law of large numbers.This result was extended in [4] to measures with unbounded support.Continuing the study of limiting distributions in non-commutative probability, we provide in this article some new limiting laws related to the free multiplicative law of large numbers and involving free and Boolean additive convolutions.For the clarity of the presentation of our results, we need to first recall some concepts of importance in non-commutative probability.
Denote by P (respectively, P + ) the set of probability measures on R (respectively, R + ).The Cauchy-Stieltjes transform G µ (.) of µ ∈ P is defined, for y ∈ C \ supp(µ), as where supp(µ) denotes the support of the measure µ.
The free additive convolution of µ and ν ∈ P, denoted by µ ⊞ ν, is defined by where the free cumulant transform, R µ , of µ is given by , for all ξ in an appropriate domain. (3) See [5] for more details about the free cumulant transform.
A measure µ ∈ P is ⊞-infinitely divisible if for each q ∈ N, there exists µ q ∈ P such that µ = µ q ⊞ ..... ⊞ µ q q times .Denote by µ ⊞t the t-fold free additive convolution of µ with itself.This operation is well defined for all t ≥ 1, (see [6]) and A measure µ ∈ P is ⊞-infinitely divisible if µ ⊞t is well defined for all t > 0.
The Boolean additive convolution is another interesting convolution in the theory of non-commutative probability, see [7].For µ, ν ∈ P, the Boolean additive convolution µ ⊎ ν is the probability measure defined by where denotes the Boolean cumulant transform of the measure µ.
We come now to the concept of free multiplicative convolution.For µ ∈ P + , (µ ̸ = δ 0 ), the S-transform is given by for all ζ in a neighborhood of 0.
In [9] (Theorem 3.1), an interesting description is given for the free multiplicative law of large numbers Φ(σ) in terms of the pseudo-variance function V σ (.) of the Cauchy-Stieltjes kernel (CSK) family generated by σ (see the next section for CSK families and the corresponding pseudo-variance functions).A number of explicit examples are given for Φ(σ), see [9].

Cauchy-Stieltjes Kernel Families
We introduce some preliminaries about CSK families and their corresponding pseudovariance functions, see [10] for more details.
Let µ be a probability measure which is non-degenerate with support bounded from above.The transform The one-sided CSK family generated by µ is the set . This gives a re-parametrization (by the mean) of K + (µ).Consider ψ µ (.), the inverse of k µ (.), and for m ∈ (m (Here, 1/0 is interpreted as ∞.)It was proven in [11] that If the support of µ is bounded from below, one may similarly introduce the one-sided CSK family.Denote this family by K − (µ).We have The variance function is (see [12]) If µ does not have the first moment, all the laws that belong to the CSK family generated by µ have infinite variance.The concept of the pseudo-variance function V µ (.) was introduced in [11].It is defined by If Throughout the following two remarks, we recall some facts that will be used in the proof of the main result of the paper given by Theorem 2.
Remark 1. (i) Consider φ : ξ −→ λξ + β, where λ ̸ = 0 and β ∈ R, and let φ(µ) be the image of µ by φ.Then, for all x close enough to m (ii) According to [11] (Proposition 3.10), for all t > 0 such that µ ⊞t is defined and for all x close enough to m (iii) We know from [13] that for all r > 0 and for all x close enough to m Furthermore, for all s ≥ 0 and for all x close enough to m Remark 2. For t > 0 such that µ ⊞t is defined, there exists an injective analytic map w t : and H t (w t (ξ)) = ξ, where For more details about the subordination function, see [14] (Theorem 2.5).