Studies on Cauchy–Stieltjes Kernel Families

Abstract

In the setting of Cauchy–Stieltjes kernel (CSK) families, this study provides some features of free Poisson, free Gamma, and free Binomial laws, as well as some innovative limit theorems linked to Fermi convolution. These findings highlight the fundamental links between noncommutative probability and analytic function theory, demonstrating the usefulness of CSK families for advancing the computational and theoretical aspects of free harmonic analysis.

1. Introduction

Free harmonic analysis, a fundamental area within free probability theory, generalizes classical harmonic analysis to noncommutative probability spaces. Arising from the study of non-commuting random variables, it offers powerful analytic tools for investigating the spectral behavior of large random matrices and operator-valued phenomena. Key constructs such as the R-transform and S-transform serve as noncommutative counterparts to the classical Fourier and Laplace transforms, enabling the study of convolutions and asymptotic distributions in free probabilistic settings. While significant theoretical developments have been made, bridging these abstract concepts with computational methods remains a challenge particularly for intricate free convolutions. Studying free harmonic analysis not only deepens our understanding of non-commutative probability but also provides powerful techniques for addressing longstanding problems in operator algebras and mathematical physics. Its tools have far-reaching applications, including in the development of free entropy, free convolution, and spectral distributions of random matrices. As such, free harmonic analysis is more than a theoretical pursuit: it is a key to unlocking insights into the structure and dynamics of systems governed by non-commutative laws, offering a rich and promising landscape for both pure and applied mathematical research as presented in [1,2,3].

However, a central task in free harmonic analysis, probability theory and statistics is to determine and characterize the properties of probability measures, as this underpins the understanding of random phenomena and their distributions. Insight into these properties enables researchers to draw inferences about stochastic processes, develop novel statistical methodologies, and tackle applied problems such as estimation, hypothesis testing, and model selection as presented in [4,5,6,7]. On the other hand, limit theorems play a foundational role in probability theory by describing the asymptotic behavior of sequences of random variables. In the context of free harmonic analysis, these theorems are equally crucial, providing insight into the convergence of noncommutative random variables and the emergent spectral distributions in large-scale systems such as random matrices; see [8,9,10,11]. By characterizing how noncommutative structures behave under aggregation, limit theorems form a cornerstone for both advancing theory and informing applications in free harmonic analysis. In this context, the study of Cauchy–Stieltjes kernel (CSK) families of probability measures holds particular significance. Defined via the Cauchy–Stieltjes transform, these families offer a robust analytic framework for describing distributions. A key feature of this framework is the relative variance function (VF), which encapsulates essential structural properties of the measures and governs the behavior of various statistical quantities, including estimators and fluctuations. Investigating the behavior and structure of these VFs not only enhances theoretical understanding but also supports the development of effective computational and analytical tools.

In the framework of CSK families, the purpose of this paper is twofold. On the one hand, it seeks to contribute to the theoretical foundations of free harmonic analysis; on the other hand, it develops computational tools that enhance our ability to model and analyze systems arising in free probability. Within this context, the study examines several fundamental free probability laws, namely the free Poisson (FP), free Gamma (FG), and free Binomial (FB) distributions. It also introduces new limit theorems associated with the Fermi convolution, highlighting the interplay between free and Boolean additive structures. To present these results in a coherent way, it is necessary to begin with a review of CSK families and their VFs. These families of probability measures are constructed in analogy with classical natural exponential families (NEFs). The key difference lies in the kernel: the exponential kernel $exp\left(\vartheta \zeta \right)$ used in NEFs is replaced, in the CSK setting, by the rational kernel $1/(1-\vartheta \zeta )$. The literature has already established important groundwork on CSK families. In particular, compactly supported CSK families were studied in [12,13], while [14] extended the theory to measures with one-sided support boundaries. These contributions provide the foundation on which the present work builds, allowing us to explore both structural properties and novel asymptotic behaviors within the CSK framework.

Let $\mathcal{P}$ denote the set of all non-degenerate probability measures on $\mathbb{R}$, and let ${\mathcal{P}}_{ba}\subset \mathcal{P}$ be the subset consisting of those probability measures whose support admits a one-sided boundary from above. For any $\rho \in {\mathcal{P}}_{ba}$, define

$$1/{\vartheta }_{+}^{\rho }=max\{0,sup\mathrm{supp}\left(\rho \right)\}.$$

The transform

$${\mathbb{M}}_{\rho }\left(\vartheta \right)=\int {(1-\vartheta \zeta )}^{-1}\rho \left(d\zeta \right)$$

is finite for every $\vartheta \in [0,{\vartheta }_{+}^{\rho })$.

The CSK family (of probability measures) associated with $\rho$ is then defined as

$${\mathbb{F}}_{+}\left(\rho \right)=\left\{{\mathbb{P}}_{\vartheta }^{\rho }\left(d\zeta \right)={\displaystyle \frac{\rho \left(d\zeta \right)}{{\mathbb{M}}_{\rho }\left(\vartheta \right)(1-\vartheta \zeta )}}:\vartheta \in (0,{\vartheta }_{+}^{\rho })\right\}.$$

The mean of ${\mathbb{P}}_{\vartheta }^{\rho }$ is given by

$${\mathbb{K}}_{\rho }\left(\vartheta \right)=\int \zeta {\mathbb{P}}_{\vartheta }^{\rho }\left(d\zeta \right),$$

and the mapping $\vartheta \mapsto {\mathbb{K}}_{\rho }\left(\vartheta \right)$ is bijective from $(0,{\vartheta }_{+}^{\rho })$ onto the interval

$$(m_1^{\rho },m_{+}^{\rho })={\mathbb{K}}_{\rho }\left((0,{\vartheta }_{+}^{\rho })\right).$$

This interval is called the mean domain of ${\mathbb{F}}_{+}\left(\rho \right)$; see [14].

A mean-parametrization of ${\mathbb{F}}_{+}\left(\rho \right)$ can be introduced as follows. Denote the inverse of ${\mathbb{K}}_{\rho }(·)$ by ${\Phi }_{\rho }(·)$. For each $m\in (m_1^{\rho },m_{+}^{\rho })$ define

$${\mathbb{Q}}_m^{\rho }\left(d\zeta \right)={\mathbb{P}}_{{\Phi }_{\rho }\left(m\right)}^{\rho }\left(d\zeta \right).$$

With this notation,

$${\mathbb{F}}_{+}\left(\rho \right)=\{{\mathbb{Q}}_m^{\rho }\left(d\zeta \right):m\in (m_1^{\rho },m_{+}^{\rho })\}.$$

The endpoints of the mean domain are given explicitly by

$$m_1^{\rho }=\underset{\vartheta \to 0^{+}}{lim}{\mathbb{K}}_{\rho }\left(\vartheta \right),m_{+}^{\rho }={\mathbb{B}}_{\rho }-\underset{w\to {\mathbb{B}}_{\rho }^{+}}{lim}1/{\mathbb{G}}_{\rho }\left(w\right),$$

where ${\mathbb{B}}_{\rho }=1/{\vartheta }_{+}^{\rho }$. Here,

$${\mathbb{G}}_{\rho }\left(w\right)=\int {\displaystyle \frac{\rho \left(dx\right)}{w-x}},w\in \mathbb{C}\setminus \mathrm{supp}\left(\rho \right)$$

is the Cauchy–Stieltjes transform (CST) of $\rho$.

If the support of $\rho$ is bounded from below, the corresponding CSK family is denoted by ${\mathbb{F}}_{-}\left(\rho \right)$. In this case, the parameter $\vartheta$ ranges over the interval $({\vartheta }_{-}^{\rho },0)$, where ${\vartheta }_{-}^{\rho }$ is equal either to $1/{\mathbb{A}}_{\rho }$ or to $-\infty$. Here, ${\mathbb{A}}_{\rho }=min\{0,infsupp\left(\rho \right)\}$. The associated domain of means for ${\mathbb{F}}_{-}\left(\rho \right)$ is the interval $(m_{-}^{\rho },m_1^{\rho })$, where

$$m_{-}^{\rho }={\mathbb{A}}_{\rho }-1/{\mathbb{G}}_{\rho }\left({\mathbb{A}}_{\rho }\right).$$

If, in addition, the support of $\rho$ is compact, one can define the two-sided CSK family as

$$\mathbb{F}\left(\rho \right)={\mathbb{F}}_{+}\left(\rho \right)\cup {\mathbb{F}}_{-}\left(\rho \right)\cup \left\{\rho \right\}.$$

The VF

$$m\mapsto {\mathfrak{V}}_{\rho }\left(m\right)=\int {(\zeta -m)}^2{\mathbb{Q}}_m^{\rho }\left(d\zeta \right)$$

plays a central role in the theory of CSK families as noted in [13]. Intuitively, this function describes how the variability of the distributions in the family depends on their mean parameter. However, the existence of the VF is not always guaranteed. In particular, when the first moment of $\rho \in {\mathcal{P}}_{ba}$ does not exist, every measure in the associated family ${\mathbb{F}}_{+}\left(\rho \right)$ necessarily has infinite variance.

To address this difficulty, Ref. [14] introduced the notion of the pseudo-variance function (PVF). This generalized concept is defined by

$${\mathbb{V}}_{\rho }\left(m\right)=m\left({\displaystyle \frac{1}{{\Phi }_{\rho }\left(m\right)}}-m\right).$$

The PVF provides a meaningful way to capture variability even in cases where the ordinary VF is not well defined.

Furthermore, if the mean of $\rho$, denoted by $m_1^{\rho }=\int \zeta \rho \left(d\zeta \right)$, is finite, then the VF ${\mathfrak{V}}_{\rho }(·)$ exists. In this case, the relationship between the VF and the PVF is made precise in [14] by the formula

$${\mathbb{V}}_{\rho }\left(m\right)={\displaystyle \frac{m}{m-m_1^{\rho }}}{\mathfrak{V}}_{\rho }\left(m\right).$$

(1)

This connection shows that the PVF extends the concept of VF in a consistent manner, covering both the finite-variance and infinite-variance regimes.

The free and Boolean additive convolutions will play an crucial role in this paper. The additive free convolution of $\lambda$ and $\varrho \in \mathcal{P}$ is the measure denoted $\lambda ⊞\varrho$ satisfying

$${\mathfrak{R}}_{\lambda ⊞\varrho }\left(\xi \right)={\mathfrak{R}}_{\lambda }\left(\xi \right)+{\mathfrak{R}}_{\varrho }\left(\xi \right),$$

where ${\mathfrak{R}}_{\lambda }$ denotes the free cumulant transform of $\lambda$ and is introduced as [15]

$${\mathfrak{R}}_{\lambda }\left({\mathbb{G}}_{\lambda }\left(\xi \right)\right)=\xi -1/{\mathbb{G}}_{\lambda }\left(\xi \right),\xi \mathrm{large}\mathrm{enough}.$$

A measure $\lambda \in \mathcal{P}$ is infinitely divisible with respect to ⊞ if for each $k\in \mathbb{N}$, ${\lambda }_k\in \mathcal{P}$ exists so that

$$\lambda =\underset{k\mathrm{times}}{\underbrace{{\lambda }_k⊞.....⊞{\lambda }_k}}.$$

The s-fold free additive convolution of $\lambda$ with itself is denoted by ${\lambda }^{⊞s}$. It is well defined for $s\ge 1$ and [16]

$${\mathfrak{R}}_{{\lambda }^{⊞s}}\left(w\right)=s{\mathfrak{R}}_{\lambda }\left(w\right).$$

A probability measure $\lambda \in \mathcal{P}$ is infinitely divisible with respect to ⊞ if ${\lambda }^{⊞s}$ is well defined for all $s>0$.

The Boolean additive convolution $\sigma \uplus \rho$ of $\sigma$ and $\rho \in \mathcal{P}$ is the measure provided by

$${\mathbf{E}}_{\sigma \uplus \rho }\left(\xi \right)={\mathbf{E}}_{\sigma }\left(\xi \right)+{\mathbf{E}}_{\rho }\left(\xi \right),$$

where

$${\mathbf{E}}_{\sigma }\left(\xi \right)=\xi -1/{\mathbb{G}}_{\sigma }\left(\xi \right),$$

is the Boolean cumulant transform of $\sigma$.

A probability measure $\sigma \in \mathcal{P}$ is infinitely divisible with respect to ⊎ if for every $p\in \mathbb{N}$, ${\sigma }_p\in \mathcal{P}$ exists so that

$$\sigma =\underset{ptimes}{\underbrace{{\sigma }_p\uplus .....\uplus {\sigma }_p}}.$$

All measures $\sigma \in \mathcal{P}$ are ⊎-infinitely divisible; see [17] (Theorem 3.6).

Now, we discuss in more details the goals of this paper: Let ${\mathbb{F}}_{+}\left(\sigma \right)=\{{\mathbb{Q}}_m^{\sigma }\left(d\zeta \right);m\in (m_1^{\sigma },m_{+}^{\sigma })\}$ be the CSK family induced by $\sigma \in {\mathcal{P}}_{ba}$, with finite moment of order 1. For $\alpha >0$, introduce the sets of measures

$${\left({\mathbb{F}}_{+}\left(\sigma \right)\right)}^{\uplus \alpha }=\left\{{\left({\mathbb{Q}}_m^{\sigma }\right)}^{\uplus \alpha }\left(d\zeta \right):m\in (m_1^{\sigma },m_{+}^{\sigma })\right\}.$$

(2)

$$D_{1/\alpha }\left({\left({\mathbb{F}}_{+}\left(\sigma \right)\right)}^{\uplus \alpha }\right)=\left\{D_{1/\alpha }\left({\left({\mathbb{Q}}_m^{\sigma }\right)}^{\uplus \alpha }\right)\left(d\zeta \right):m\in (m_1^{\sigma },m_{+}^{\sigma })\right\}.$$

(3)

In Section 2, for $\alpha >1$, we show that if ${\mathbb{F}}_{+}\left(D_{1/\alpha }\left({\sigma }^{⊞\alpha }\right)\right)={\left({\mathbb{F}}_{+}\left(\sigma \right)\right)}^{\uplus \alpha }$ (or $D_{1/\alpha }$$\left({\left({\mathbb{F}}_{+}\left(\sigma \right)\right)}^{\uplus \alpha }\right)={\mathbb{F}}_{+}\left({\sigma }^{⊞\alpha }\right)$), then $\sigma$ is a FG distribution up to a scale transformation. Here $D_c\left(\sigma \right)$ denotes the dilation of measure $\sigma$ by a number $c\ne 0$. In Section 3, we focus on the FB CSK family and establish a property that is directly linked to its location parameter. This analysis also allows us to demonstrate an important structural limitation, namely that no CSK family can be constructed on the basis of a scale parameter. Section 4 is devoted to the FP CSK family. In this part, we develop an estimation procedure for its elements, which is derived by exploiting the framework of free additive convolution. This provides a concrete tool for understanding how the elements of the FP CSK family behave under free summation. Finally, in Section 5, we turn our attention to the Fermi convolution. After presenting the necessary preliminaries, we establish several new limit theorems in this setting. These results are obtained by means of VFs and rely on a combination of both free and Boolean additive convolutions. Together, they illustrate how the interplay between different notions of non-commutative independence enriches the study of CSK families.

To close this part, some useful facts are presented through the next two remarks to aid in the proof of the article’s main results.

Remark 1. 

Let $\rho \in {\mathcal{P}}_{ba}$.

(i) 

According to [14] (Proposition 3.5), the PVF ${\mathbb{V}}_{\rho }(·)$ determine ρ: Consider $\Delta =\Delta \left(m\right)=m+{\displaystyle \frac{{\mathbb{V}}_{\rho }\left(m\right)}{m}},$ then

$${\mathbb{G}}_{\rho }(\Delta )={\displaystyle \frac{m}{{\mathbb{V}}_{\rho }\left(m\right)}}.$$

(4)

(ii) 

Let $\chi :\zeta \mapsto \kappa \zeta +\beta$ with $\beta \in \mathbb{R}$ and $\kappa \ne 0$. Then, according to [14] (Section 3.3), for m close sufficiently to $m_1^{\chi \left(\rho \right)}=\chi \left(m_1^{\rho }\right)=\kappa m_1^{\rho }+\beta$,

$${\mathbb{V}}_{\chi \left(\rho \right)}\left(m\right)={\displaystyle \frac{{\kappa }^{2}m}{m-\beta }}{\mathbb{V}}_{\rho }\left({\displaystyle \frac{m-\beta }{\kappa }}\right).$$

(5)

If the VF exists,

$${\mathfrak{V}}_{\chi \left(\rho \right)}\left(m\right)={\kappa }^2{\mathfrak{V}}_{\rho }\left({\displaystyle \frac{m-\beta }{\kappa }}\right).$$

(6)

(iii) 

For $\alpha >0$, so that ${\rho }^{⊞\alpha }$ is defined and for m close sufficiently to $m_1^{{\rho }^{⊞\alpha }}=\alpha m_1^{\rho }$, we have [14]

$${\mathbb{V}}_{{\sigma }^{⊞\alpha }}\left(m\right)=\alpha {\mathbb{V}}_{\sigma }(m/\alpha ).$$

(7)

If the VF exists,

$${\mathfrak{V}}_{{\rho }^{⊞\alpha }}\left(m\right)=\alpha {\mathfrak{V}}_{\rho }(m/\alpha ).$$

(8)

(iv) 

For $\alpha >0$ and for m close enough to $m_1^{{\rho }^{\uplus \alpha }}=\alpha m_1^{\rho }$, we have [18]

$${\mathbb{V}}_{{\sigma }^{\uplus \alpha }}\left(m\right)=m^2(1/\alpha -1)+\alpha {\mathbb{V}}_{\sigma }(m/\alpha ).$$

(9)

If $m_1^{\rho }$ is finite,

$${\mathfrak{V}}_{{\rho }^{\uplus \alpha }}\left(m\right)=m(m-\alpha m_1^{\rho })(1/\alpha -1)+\alpha {\mathfrak{V}}_{\rho }(m/\alpha ).$$

(10)

In particular the variance of ${\rho }^{\uplus \alpha }$ is $\mathrm{Var}\left({\rho }^{\uplus \alpha }\right)=\alpha \mathrm{Var}\left(\rho \right)$.

(v) 

From [14] (Corollary 3.6), we have that $m^2/{\mathbb{V}}_{\rho }\left(m\right)\stackrel{m\searrow m_1^{\rho }}{\to }0.$

Remark 2. 

For $\rho \in {\mathcal{P}}_{ba}$, it is well known that the mean $m_1^{\rho }$ exhibits a simple behavior under both affine transformation and free additive convolution powers. Specifically, we have

$$m_1^{\chi \left(\rho \right)}=\chi \left(m_1^{\rho }\right)=\kappa m_1^{\rho }+\beta$$

and for $\alpha >0$ whenever ${\rho }^{⊞\alpha }$ is well-defined,

$$m_1^{{\rho }^{⊞\alpha }}=\alpha m_1^{\rho }.$$

In contrast, no general formula is available for the upper boundary of the mean domain, $m_{+}^{\rho }$, when considering either affine transformations or free additive convolution powers of ρ. To address this limitation, the authors in [19] extended the notion of the mean domain in a way that preserves the PVF. In particular, they defined the upper end of the extended mean domain as

$${\mathit{M}}_{+}^{\rho }=inf\left\{m>m_1^{\rho }:{\mathbb{V}}_{\rho }\left(m\right)/m<0\right\}.$$

As shown in [19] (Section 3.2), this extended boundary behaves well under free additive convolution powers: for all $\alpha >0$ so that ${\rho }^{⊞\alpha }$ is defined, one has

$${\mathit{M}}_{+}^{{\rho }^{⊞\alpha }}=\alpha {\mathit{M}}_{+}^{\rho }.$$

Similarly, under dilation, we have

$${\mathit{M}}_{+}^{D_{1/\alpha }\left(\rho \right)}={\mathit{M}}_{+}^{\rho }/\alpha .$$

Accordingly, in Section 2, for $\rho \in {\mathcal{P}}_{ba}$, we will focus on the mean domain of the form $(m_1^{\rho },{\mathit{M}}_{+}^{\rho })$.

2. Notes on the FG Law

For $a\ne 0$, the FG law is given by

$${\mathbf{fg}}_{\left(a\right)}\left(d\zeta \right)={\displaystyle \frac{\sqrt{({(\sqrt{a^2+1}+a)}^2-\zeta )(\zeta -{(\sqrt{a^2+1}-a)}^2)}}{2\pi a^2{\zeta }^2}}{\mathbf{1}}_{\left((\sqrt{a^2+1}-{\left|a\right|)}^2,(\sqrt{a^2+1}+{\left|a\right|)}^2\right)}\left(\zeta \right)d\zeta .$$

(11)

We have

$${\mathfrak{V}}_{{\mathbf{fg}}_{\left(a\right)}}\left(m\right)=a^2{m}^2\mathrm{and}(m_1^{{\mathbf{fg}}_{\left(a\right)}},{\mathbf{M}}_{+}^{{\mathbf{fg}}_{\left(a\right)}})=(1,+\infty ).$$

(12)

The FG law represents a fundamental concept in the framework of free probability. It serves as the non-commutative analogue of the classical Gamma distribution, which occupies a central role in traditional probability theory. In this sense, the FG law provides a powerful tool for modeling classes of random variables that arise naturally in the free setting. Understanding its structure and properties is therefore essential for analyzing the distributional behavior of free random variables. The importance of the FG law extends to several areas of free probability and its applications, including random matrix theory, additive and multiplicative free convolutions, and non-commutative statistical models. Characterizing this distribution helps to shed light on the underlying mechanisms that govern these phenomena. Over the past few decades, the FG law has received considerable attention in the literature. For instance, the work of Haagerup and Thorbjørnsen [20] has highlighted several significant properties of the density of the FG law. Among these are the asymptotic behavior of the distribution, its unimodality, and its analyticity, which together form a comprehensive description of its analytical profile. From another perspective, Bryc [13] has shown that the FG law can be characterized through its VF, and that it appears naturally as a member of the free Meixner family. This characterization links the FG law with a broader class of well-structured distributions in free probability. In the present section, we aim to build upon these established results. More precisely, we expand the understanding of the FG law by presenting additional properties and insights that further clarify its role in the landscape of free distributions. To be specific, we establish the following results:

Theorem 1. 

Let $\sigma \in {\mathcal{P}}_{ba}$, with finite first moment $m_1^{\sigma }$. For $\alpha >1$, consider the sets of measures defined by (2) and (3). If

(i) 

${\left({\mathbb{F}}_{+}\left(\sigma \right)\right)}^{\uplus \alpha }={\mathbb{F}}_{+}\left(D_{1/\alpha }\left({\sigma }^{⊞\alpha }\right)\right)$ or

(ii) 

$D_{1/\alpha }\left({\left({\mathbb{F}}_{+}\left(\sigma \right)\right)}^{\uplus \alpha }\right)={\mathbb{F}}_{+}\left({\sigma }^{⊞\alpha }\right)$

then σ is a FG law (11) up to a scale transformation.

Proof. 

(i) Suppose that ${\left({\mathbb{F}}_{+}\left(\sigma \right)\right)}^{\uplus \alpha }={\mathbb{F}}_{+}\left(D_{1/\alpha }\left({\sigma }^{⊞\alpha }\right)\right)$. Then, ∀$t\in (m_1^{\sigma },{\mathbf{M}}_{+}^{\sigma })$, there exists $r\in (m_1^{D_{1/\alpha }\left({\sigma }^{⊞\alpha }\right)},{\mathbf{M}}_{+}^{D_{1/\alpha }\left({\sigma }^{⊞\alpha }\right)})=(m_1^{\sigma },{\mathbf{M}}_{+}^{\sigma })$ so that

$${\left({\mathbb{Q}}_t^{\sigma }\right)}^{\uplus \alpha }={\mathbb{Q}}_r^{D_{1/\alpha }\left({\sigma }^{⊞\alpha }\right)}.$$

(13)

That is,

$${\mathfrak{R}}_{{\left({\mathbb{Q}}_t^{\sigma }\right)}^{\uplus \alpha }}\left(\xi \right)={\mathfrak{R}}_{{\mathbb{Q}}_r^{D_{1/\alpha }\left({\sigma }^{⊞\alpha }\right)}}\left(\xi \right),\xi \mathrm{close}\mathrm{to}0.$$

(14)

As $m_1^{\sigma }$ is finite, from [21], we know that

$${\mathfrak{R}}_{{\left({\mathbb{Q}}_t^{\sigma }\right)}^{\uplus \alpha }}\left(\xi \right)=m_1^{{\left({\mathbb{Q}}_t^{\sigma }\right)}^{\uplus \alpha }}+\mathrm{Var}\left({\left({\mathbb{Q}}_t^{\sigma }\right)}^{\uplus \alpha }\right)\xi +\xi \epsilon \left(\xi \right)=\alpha t+\alpha {\mathfrak{V}}_{\sigma }\left(t\right)\xi +\alpha \xi \epsilon \left(\xi \right),\epsilon \left(\xi \right)\stackrel{\xi \to 0}{\to }0.$$

(15)

Basing on (6) and (8), we may write according to [21]

$${\mathfrak{R}}_{{\mathbb{Q}}_r^{D_{1/\alpha }\left({\sigma }^{⊞\alpha }\right)}}\left(\xi \right)=r+{\mathfrak{V}}_{D_{1/\alpha }\left({\sigma }^{⊞\alpha }\right)}\left(r\right)\xi +\xi {\epsilon }_1\left(\xi \right)=r+{\displaystyle \frac{1}{\alpha }}{\mathfrak{V}}_{\sigma }\left(r\right)\xi +\xi {\epsilon }_1\left(\xi \right),{\epsilon }_1\left(\xi \right)\stackrel{\xi \to 0}{\to }0.$$

(16)

Combining (15) and (16), relation (14) becomes

$$\alpha t+\alpha {\mathfrak{V}}_{\sigma }\left(t\right)\xi +\alpha \xi \epsilon \left(\xi \right)=r+{\displaystyle \frac{1}{\alpha }}{\mathfrak{V}}_{\sigma }\left(r\right)\xi +\xi {\epsilon }_1\left(\xi \right).$$

(17)

From (17), one see that $r=\alpha t$ and then

$${\mathfrak{V}}_{\sigma }\left(\alpha t\right)={\alpha }^2{\mathfrak{V}}_{\sigma }\left(t\right),\forall t\in (m_1^{\sigma },{\mathbf{M}}_{+}^{\sigma })\mathrm{and}\forall \alpha >1.$$

(18)

${\mathfrak{V}}_{\sigma }(·)\ne 0$ as $\sigma$ is non-degenerate. Thus, relation (18) gives that ${\mathfrak{V}}_{\sigma }\left(t\right)=\lambda t^2$ for some $\lambda >0$.

• If $m_1^{\sigma }=0$, then there is no VF of the form $V\left(t\right)=\lambda t^2$, with $\lambda >0$. See [22].
• If $m_1^{\sigma }\ne 0$, then $\sigma$ is the image by $\zeta \mapsto m_1^{\sigma }\zeta$ of the FG law (11) and we have $\lambda =a^2$.

□

Remark 3. 

Suppose that $m_1^{\sigma }>0$. In that case, we have $(m_1^{\sigma },{\mathit{M}}_{+}^{\sigma })=(m_1^{\sigma },+\infty ).$ Then for $\alpha >1$, if $t\in (m_1^{\sigma },+\infty )$, we have $r=\alpha t\in (\alpha m_1^{\sigma },+\infty )\subset (m_1^{\sigma },+\infty )$. Relation (13) is hence clearly stated. If $m_1^{\sigma }<0$, an identical result is reached in this instance, where the one-sided domain of means is $(-\infty ,m_1^{\sigma })$.

The inverse implication of (i) is not valid. Suppose that $m_1^{\sigma }>0$ and $\sigma$ is the image by $\zeta \mapsto m_1^{\sigma }\zeta$ of ${\mathbf{fg}}_{\left(a\right)}$. We show that

$${\left({\mathbb{Q}}_t^{\sigma }\right)}^{\uplus \alpha }\ne {\mathbb{Q}}_{\alpha t}^{D_{1/\alpha }\left({\sigma }^{⊞\alpha }\right)}.$$

(19)

We have that $m_1^{{\left({\mathbb{Q}}_t^{\sigma }\right)}^{\uplus \alpha }}=\alpha t=m_1^{{\mathbb{Q}}_{\alpha t}^{D_{1/\alpha }\left({\sigma }^{⊞\alpha }\right)}}$. Then, there is $\varsigma >0$ so that ${\mathbb{V}}_{{\left({\mathbb{Q}}_t^{\sigma }\right)}^{\uplus \alpha }}(·)$ and ${\mathbb{V}}_{{\mathbb{Q}}_{\alpha t}^{D_{1/\alpha }\left({\sigma }^{⊞\alpha }\right)}}(·)$ are well defined on $(\alpha t,\alpha t+\varsigma )$. We know from [22] (Equation (41)) that

$${\mathbb{V}}_{{\mathbb{Q}}_t^{\sigma }}\left(y\right)={\displaystyle \frac{y^3(a^2{t}^2-(y-t)(t-m_1^{\sigma }))}{(y-t)(y(t-m_1^{\sigma })+t{m}_1^{\sigma })}},y\ne t.$$

(20)

Combining (40) and (9), we obtain

$${\mathbb{V}}_{{\left({\mathbb{Q}}_t^{\sigma }\right)}^{\uplus \alpha }}\left(y\right)={\displaystyle \frac{y^3(a^2{t}^2-(y-t)(t-m_1^{\sigma }))}{(y-t)(y(t-m_1^{\sigma })+t{m}_1^{\sigma })}}+y^2(1/\alpha -1),y\ne t.$$

(21)

Now, we calculate ${\mathbb{V}}_{{\mathbb{Q}}_r^{D_{1/\alpha }\left({\sigma }^{⊞\alpha }\right)}}(·)$. Basing on (7), (5) and (12), for $u\in (m_1^{\sigma },+\infty )$, we have

$${\mathbb{V}}_{D_{1/\alpha }\left({\sigma }^{⊞\alpha }\right)}\left(u\right)={\displaystyle \frac{a^2{u}^3}{\alpha (u-m_1^{\sigma })}}.$$

(22)

Using [19] (Equation (2.9)), we obtain

$$z={\displaystyle \frac{u^2\left(\frac{a^2{r}^3}{\alpha (r-m_1^{\sigma })}\right)-r^2\left(\frac{a^2{u}^3}{\alpha (u-m_1^{\sigma })}\right)}{u\left(\frac{a^2{r}^3}{\alpha (r-m_1^{\sigma })}\right)-r\left(\frac{a^2{u}^3}{\alpha (u-m_1^{\sigma })}\right)}}={\displaystyle \frac{ur{m}_1^{\sigma }}{m_1^{\sigma }(u+r)-ur}}.$$

(23)

Equation (23) gives

$$u={\displaystyle \frac{zr{m}_1^{\sigma }}{zr+r{m}_1^{\sigma }-z{m}_1^{\sigma }}}$$

(24)

Using [19] (Equation (2.10)), (22) and (24), we obtain

$${\mathbb{V}}_{{\mathbb{Q}}_r^{D_{1/\alpha }\left({\sigma }^{⊞\alpha }\right)}}\left(z\right)=z\left({\displaystyle \frac{{\mathbb{V}}_{D_{1/\alpha }\left({\sigma }^{⊞\alpha }\right)}\left(u\right)}{u}}+u-z\right)={\displaystyle \frac{z^3(a^2{r}^2-\alpha (z-r)(r-m_1^{\sigma }))}{\alpha (z-r)(z(r-m_1^{\sigma })+r{m}_1^{\sigma })}},z\ne r.$$

(25)

Equations (25) and (21) give

$${\mathbb{V}}_{{\mathbb{Q}}_r^{D_{1/\alpha }\left({\sigma }^{⊞\alpha }\right)}}\left(x\right)\ne {\mathbb{V}}_{{\left({\mathbb{Q}}_t^{\sigma }\right)}^{\uplus \alpha }}\left(x\right),\forall x\in (\alpha t,\alpha t+\varsigma ).$$

(26)

This ends the proof of (19) based on (26) and (4).

(ii) Assume that ${\mathbb{F}}_{+}\left({\sigma }^{⊞\alpha }\right)=D_{1/\alpha }\left({\left({\mathbb{F}}_{+}\left(\sigma \right)\right)}^{\uplus \alpha }\right)$. Then, ∀$t\in (m_1^{{\sigma }^{⊞\alpha }},{\mathbf{M}}_{+}^{{\sigma }^{⊞\alpha }})=(\alpha m_1^{\sigma },\alpha {\mathbf{M}}_{+}^{\sigma })$, there exists $s\in (m_1^{\sigma },{\mathbf{M}}_{+}^{\sigma })$ so that

$${\mathbb{Q}}_t^{{\sigma }^{⊞\alpha }}=D_{1/\alpha }\left({\left({\mathbb{Q}}_s^{\sigma }\right)}^{\uplus \alpha }\right).$$

(27)

That is, for $\xi$ close to 0

$${\mathfrak{R}}_{{\mathbb{Q}}_t^{{\sigma }^{⊞\alpha }}}\left(\xi \right)={\mathfrak{R}}_{D_{1/\alpha }\left({\left({\mathbb{Q}}_s^{\sigma }\right)}^{\uplus \alpha }\right)}\left(\xi \right).$$

(28)

Equation (28) may be expressed as

$$t+\alpha {\mathfrak{V}}_{\sigma }(t/\alpha )\xi +\xi \epsilon \left(\xi \right)=s+{\mathfrak{V}}_{\sigma }\left(s\right){\displaystyle \frac{\xi }{\alpha }}+\xi {\epsilon }_2\left(\xi \right),\epsilon \left(\xi \right)\stackrel{\xi \to 0}{\to }0\mathrm{and}{\epsilon }_2\left(\xi \right)\stackrel{\xi \to 0}{\to }0.$$

(29)

Since $s=t$ is evident from (29), ${\mathfrak{V}}_{\sigma }(t/\alpha )={\mathfrak{V}}_{\sigma }\left(t\right)/{\alpha }^2$, ∀$t\in (\alpha m_1^{\sigma },\alpha {\mathbf{M}}_{+}^{\sigma })$, and ∀$\alpha >1$. The identical conclusion is given as in (i): In other words, $m_1^{\sigma }\ne 0$ and $\sigma$ is the image of the FG law (11) by $\zeta \mapsto m_1^{\sigma }\zeta$.

Remark 4. 

Suppose that $m_1^{\sigma }>0$. In that case, we have $(m_1^{{\sigma }^{⊞\alpha }},{\mathit{M}}_{+}^{{\sigma }^{⊞\alpha }})=(\alpha m_1^{\sigma },+\infty )$. Then $\forall \alpha >1$, if $t\in (\alpha m_1^{\sigma },+\infty )$, then $s=t\in (\alpha m_1^{\sigma },+\infty )\subset (m_1^{\sigma },+\infty )$. Thus, relation (27) is defined. The identical conclusion is made if $m_1^{\sigma }<0$.

The reverse implication of (ii) is also not valid. That is,

$$D_{1/\alpha }\left({\left({\mathbb{Q}}_t^{\sigma }\right)}^{\uplus \alpha }\right)\ne {\mathbb{Q}}_t^{\left({\sigma }^{⊞\alpha }\right)}.$$

(30)

We have that $m_1^{D_{1/\alpha }\left({\left({\mathbb{Q}}_t^{\sigma }\right)}^{\uplus \alpha }\right)}=t=m_1^{{\mathbb{Q}}_t^{\left({\sigma }^{⊞\alpha }\right)}}$. Then, $\iota >0$ exists so that ${\mathbb{V}}_{D_{1/\alpha }\left({\left({\mathbb{Q}}_t^{\sigma }\right)}^{\uplus \alpha }\right)}(·)$ and ${\mathbb{V}}_{{\mathbb{Q}}_t^{\left({\sigma }^{⊞\alpha }\right)}}(·)$ are well defined on $(t,t+\iota )$. Basing on (5) and (21), ∀$x\in (t,t+\iota )$, we have

$${\mathbb{V}}_{D_{1/\alpha }\left({\left({\mathbb{Q}}_t^{\sigma }\right)}^{\uplus \alpha }\right)}\left(x\right)={\displaystyle \frac{1}{{\alpha }^2}}{\mathfrak{V}}_{{\left({\mathbb{Q}}_t^{\sigma }\right)}^{\uplus \alpha }}\left(\alpha x\right)={\displaystyle \frac{\alpha x^3(a^2{t}^2-(\alpha x-t)(t-m_1^{\sigma }))}{(\alpha x-t)[\alpha x(t-m_1^{\sigma })+t{m}_1^{\sigma }]}}+x^2(1/\alpha -1).$$

(31)

We calculate ${\mathbb{V}}_{{\mathbb{Q}}_t^{\left({\sigma }^{⊞\alpha }\right)}}(·)$. For $m\in (\alpha m_1^{\sigma },+\infty )$, basing on (7) and (12), we have

$${\mathbb{V}}_{{\sigma }^{⊞\alpha }}\left(m\right)=\alpha {\mathbb{V}}_{\sigma }(m/\alpha )={\displaystyle \frac{1}{\alpha }}{\displaystyle \frac{a^2{m}^3}{m-\alpha m_1^{\sigma }}}.$$

(32)

Using [19] (Equation (2.9)), we get

$$w={\displaystyle \frac{m^2\left(\frac{a^2{t}^3}{\alpha (t-\alpha m_1^{\sigma })}\right)-t^2\left(\frac{a^2{m}^3}{\alpha (m-\alpha m_1^{\sigma })}\right)}{m\left(\frac{a^2{t}^3}{\alpha (t-\alpha m_1^{\sigma })}\right)-t\left(\frac{a^2{m}^3}{\alpha (m-\alpha m_1^{\sigma })}\right)}}={\displaystyle \frac{mt\alpha m_1^{\sigma }}{\alpha m_1^{\sigma }(m+t)-mt}}.$$

(33)

Equation (33) gives

$$m={\displaystyle \frac{wt\alpha m_1^{\sigma }}{tw+\alpha m_1^{\sigma }t-\alpha m_1^{\sigma }w}}$$

(34)

Using [19] (Equation (2.10)), (32) and (34), we obtain

$${\mathbb{V}}_{{\mathbb{Q}}_t^{{\sigma }^{⊞\alpha }}}\left(w\right)=w\left({\displaystyle \frac{{\mathbb{V}}_{{\sigma }^{⊞\alpha }}\left(m\right)}{m}}+m-w\right)={\displaystyle \frac{w^3(a^2{t}^2-\alpha (w-t)(t-\alpha m_1^{\sigma }))}{\alpha (w-t)(tw-w\alpha m_1^{\sigma }+t\alpha m_1^{\sigma })}},w\ne t.$$

(35)

One sees from (31) and (35) that ${\mathbb{V}}_{{\mathbb{Q}}_t^{{\sigma }^{⊞\alpha }}}\left(x\right)\ne {\mathbb{V}}_{D_{1/\alpha }\left({\left({\mathbb{Q}}_t^{\sigma }\right)}^{⊞\alpha }\right)}\left(x\right)$, ∀$x\in (t,t+\iota )$. This ends the proof of (30).

3. Location and Scale Parameters in CSK Families

The study of the location and scale parameters in CSK families is critical because they influence the family’s basic structural aspects, similar to how classical probability works with NEFs. The location parameter indicates distribution shifts, whereas the scale parameter records dilations or contractions, both of which are important in understanding stability, invariance, and transformations during convolution processes. Analyzing these factors offers a better understanding of how CSK families operate under free probability translation and scaling, making them more useful in limit theorems, random matrix models, and free harmonic analysis. In this part, we provide a property of the FB CSK family based on the location parameter. We also demonstrate an important structural limitation for the theory of CSK families, namely that no CSK family can be constructed on the basis of a scale parameter.

According to [13] (Theorem 3.2), the FB law is given by

$$f{b}_{(a,b)}\left(dt\right)={\displaystyle \frac{\sqrt{4(1+b)-{(t-a)}^2}}{2\pi (b{t}^2+at+1)}}{1}_{(a-2\sqrt{1+b},a+2\sqrt{1+b})}\left(t\right)dt+p_1{\delta }_{t_1}+p_2{\delta }_{t_2},$$

(36)

with

$$t_{1,2}={\displaystyle \frac{-a\pm \sqrt{a^2-4b}}{2b}}\mathrm{and}{p}_{1,2}=1+{\displaystyle \frac{\sqrt{a^2-4b}\mp a}{2b\sqrt{a^2-4b}}}.$$

It induces the CSK family with $m_1^{f{b}_{(a,b)}}=0$ and VF

$${\mathfrak{V}}_{f{b}_{(a,b)}}\left(m\right)=1+am+b{m}^2,a\in \mathbb{R},-1\le b<0.$$

(37)

The parameter $\vartheta$ is said to be a location parameter if for $X_{\vartheta }\sim {\mathbb{P}}_{\vartheta }^{\rho }$, then $X_{\vartheta }\sim X+\vartheta$ with $X\sim {\mathbb{P}}_0^{\rho }=\rho$.

Theorem 2. 

Let $\rho \in {\mathcal{P}}_{ba}$. Suppose that ${\mathbb{P}}_{\vartheta }^{\rho }$ is a location family induced by ${\mathbb{P}}_0^{\rho }=\rho$, with location parameter ϑ. Then, we have the following:

(i) 

If $m_1^{\rho }=-\infty$, then there is no CSK family with location parameter.

(ii) 

If $-\infty <m_1^{\rho }$, then $\rho ={\mathbf{T}}_{m_1^{\rho }}\left(f{b}_{(-m_1^{\rho },-1)}\right)$, where ${\mathbf{T}}_{\beta }\left(x\right)=x+\beta$ with $\beta \in \mathbb{R}$ and $f{b}_{(-m_1^{\rho },-1)}$ is the FB law given by (36) with parameters $a=-m_1^{\rho }$ and $b=-1$.

Proof. 

Suppose that ${\mathbb{P}}_{\vartheta }^{\rho }$ is a location family induced by ${\mathbb{P}}_0^{\rho }=\rho$, with location parameter $\vartheta$. The $\mathfrak{R}$-transform of ${\mathbb{Q}}_m^{\rho }={\mathbb{P}}_{\vartheta }^{\rho }$ is given by,

$${\mathfrak{R}}_{{\mathbb{Q}}_m^{\rho }}\left(z\right)={\mathfrak{R}}_{{\mathbb{P}}_{\vartheta }^{\rho }}\left(z\right)={\mathfrak{R}}_{X+\vartheta }\left(z\right)={\mathfrak{R}}_{\rho }\left(z\right)+\vartheta ={\mathfrak{R}}_{\rho }\left(z\right)+{\displaystyle \frac{1}{{\mathbb{V}}_{\rho }\left(m\right)/m+m}},z\mathrm{close}\mathrm{to}0.$$

(38)

When $z\to 0$, Equation (38) gives

$$m=m_1^{\rho }+{\displaystyle \frac{1}{{\mathbb{V}}_{\rho }\left(m\right)/m+m}}.$$

(39)

From [14] (Section 2), we know that

$$-\infty \le \int t\rho \left(dt\right)=m_1^{\rho }<m_{+}^{\rho }\le supsupp\left(\rho \right)<+\infty .$$

(i) If $m_1^{\rho }=-\infty$, relation (39) gives $m=-\infty$, absurde. Then, there is no CSK family with location parameter.

(ii) If $-\infty <m_1^{\rho }$, then from (39), one can see that

$${\mathbb{V}}_{\rho }\left(m\right)={\displaystyle \frac{m}{m-m_1^{\rho }}}(1+m{m}_1^{\rho }-m^2).$$

(40)

On the other hand, based on (5) and (37), for m close to $m_1^{\rho }$, one has

$${\mathbb{V}}_{{\mathbf{T}}_{m_1^{\rho }}(f{b}_{(-m_1^{\rho },-1)})}\left(m\right)={\displaystyle \frac{m}{m-m_1^{\rho }}}(1+m{m}_1^{\rho }-m^2).$$

(41)

Equations (40) and (41) gives that ${\mathbb{V}}_{\rho }\left(m\right)={\mathbb{V}}_{{\mathbf{T}}_{m_1^{\rho }}(f{b}_{(-m_1^{\rho },-1)})}\left(m\right)$, which implies that $\rho ={\mathbf{T}}_{m_1^{\rho }}(f{b}_{(-m_1^{\rho },-1)})$ by the use of (4). □

The parameter $\vartheta$ is said a scale parameter if for $X_{\vartheta }\sim {\mathbb{P}}_{\vartheta }^{\rho }$, then $X_{\vartheta }\sim X/\vartheta$ with $X\sim \rho$.

Theorem 3. 

There is no CSK family with scale parameter.

Proof. 

Suppose that ${\mathbb{P}}_{\vartheta }^{\rho }$ is a scale family generated by $\rho$, with scale parameter $\vartheta$. For z close to 0, we have

$${\mathfrak{R}}_{{\mathbb{Q}}_m^{\rho }}\left(z\right)={\mathfrak{R}}_{{\mathbb{P}}_{\vartheta }^{\rho }}\left(z\right)={\mathfrak{R}}_{X/\vartheta }\left(z\right)={\displaystyle \frac{1}{\vartheta }}{\mathfrak{R}}_X\left({\displaystyle \frac{z}{\vartheta }}\right)=({\mathbb{V}}_{\rho }\left(m\right)/m+m){\mathfrak{R}}_{\rho }\left(({\mathbb{V}}_{\rho }\left(m\right)/m+m)z\right),$$

(42)

When $z\to 0$, Equation (42) gives,

$$m=({\mathbb{V}}_{\rho }\left(m\right)/m+m)m_1^{\rho }.$$

(43)

• If $m_1^{\rho }=-\infty$, relation (43) becomes $m=-\infty$, absurde. Then, there is no CSK family with a scale parameter.
• If $m_1^{\rho }=0$, relation (43) becomes $m=0$, absurde.
• If $m_1^{\rho }=1$, relation (43) becomes ${\mathbb{V}}_{\rho }\left(m\right)=0$ and this implies that $\rho ={\delta }_1$ which is impossible because we deal with non-degenerate measure $\rho$.
• If $m_1^{\rho }\ne -\infty ,0,1$, Equation (43) implies that

  $${\mathbb{V}}_{\rho }\left(m\right)=\left({\displaystyle \frac{1-m_1^{\rho }}{m_1^{\rho }}}\right)m^2.$$

  (44)
  But the function provided by (44) cannot serve as a PVF. We prove this fact by contraposition. Assume that $\mathbb{V}\left(m\right)=\left({\displaystyle \frac{1-m_1^{\rho }}{m_1^{\rho }}}\right)m^2$ is a PVF of a CSK family induced by a measure $\rho$ with $m_1^{\rho }\ne -\infty ,0,1$. Then ${\displaystyle \frac{m^2}{\mathbb{V}\left(m\right)}}\stackrel{m\to m_1^{\rho }}{\to }{\displaystyle \frac{m_1^{\rho }}{1-m_1^{\rho }}}\ne 0$, which contradicts Remark 1(v).

□

4. Estimation of the FP CSK Family

In free probability theory, the limit of repeated free additive convolution of the sequence of measures

$${\mu }_N=\left(1-{\displaystyle \frac{\lambda }{N}}\right){\delta }_0+{\displaystyle \frac{\lambda }{N}}{\delta }_s,\mathrm{for}s>0,N\ge 1\mathrm{and}0<\lambda <N$$

is the FP law with jump size s and rate $\lambda$. In other words,

$$\underset{N\mathrm{times}}{\underbrace{{\mu }_N⊞{\mu }_N......⊞{\mu }_N}}\stackrel{N\to +\infty }{\to }{p}_{\lambda }^{\left(s\right)}\mathrm{in}\mathrm{distribution},$$

where $p_{\lambda }^{\left(s\right)}$ is called the FP law and is given by

$$p_{\lambda }^{\left(s\right)}\left(dt\right)={\displaystyle \frac{\sqrt{4\lambda s^2-{(t-s(1+\lambda ))}^2}}{2\pi \lambda t}}{\mathbf{1}}_{(s{(1-\sqrt{\lambda })}^2,s{(1+\sqrt{\lambda })}^2)}\left(t\right)\left(dt\right)+{(1-\lambda )}^{+}{\delta }_0,$$

(45)

with $m_1^{p_{\lambda }^{\left(s\right)}}=\lambda s$. The corresponding CST is given by

$${\mathbb{G}}_{p_{\lambda }^{\left(s\right)}}\left(w\right)={\displaystyle \frac{w+s-\lambda s-\sqrt{{(w-s(1+\lambda ))}^2-4\lambda s^2}}{2sw}},w\in \mathbb{C}\setminus supp\left(p_{\lambda }^{\left(s\right)}\right).$$

(46)

The PVF of $\mathbb{F}\left(p_{\lambda }^{\left(s\right)}\right)$ is

$${\mathbb{V}}_{p_{\lambda }^{\left(s\right)}}\left(m\right)={\displaystyle \frac{s{m}^2}{m-\lambda s}}.$$

(47)

Concerning the domain of means of $\mathbb{F}\left(p_{\lambda }^{\left(s\right)}\right)$, we have the following:

• If $\lambda \le 1$, then ${\mathbb{A}}_{p_{\lambda }^{\left(s\right)}}=0$, ${\mathbb{B}}_{p_{\lambda }^{\left(s\right)}}=s{(1+\sqrt{\lambda })}^2$, $m_{-}^{p_{\lambda }^{\left(s\right)}}={\mathbb{A}}_{p_{\lambda }^{\left(s\right)}}-1/{\mathbb{G}}_{p_{\lambda }^{\left(s\right)}}({\mathbb{A}}_{p_{\lambda }^{\left(s\right)}})=0$ and $m_{+}^{p_{\lambda }^{\left(s\right)}}={\mathbb{B}}_{p_{\lambda }^{\left(s\right)}}-1/{\mathbb{G}}_{p_{\lambda }^{\left(s\right)}}({\mathbb{B}}_{p_{\lambda }^{\left(s\right)}})=s\lambda \left(1+1/\sqrt{\lambda }\right).$
• If $\lambda >1$, then ${\mathbb{A}}_{p_{\lambda }^{\left(s\right)}}=s{(1-\sqrt{\lambda })}^2$, ${\mathbb{B}}_{p_{\lambda }^{\left(s\right)}}=s{(1+\sqrt{\lambda })}^2$, $m_{-}^{p_{\lambda }^{\left(s\right)}}=s\lambda \left(1-1/\sqrt{\lambda }\right)$, and $m_{+}^{p_{\lambda }^{\left(s\right)}}=s\lambda \left(1+1/\sqrt{\lambda }\right).$

The two-sided CSK family generated $p_{\lambda }^{\left(s\right)}$ is

$$\mathbb{F}\left(p_{\lambda }^{\left(s\right)}\right)=\left\{{\mathbb{Q}}_m^{p_{\lambda }^{\left(s\right)}}\left(dx\right)={\displaystyle \frac{s{m}^2}{s{m}^2+m(m-\lambda s)(m-x)}}{p}_{\lambda }^{\left(s\right)}\left(dx\right):m\in \left(m_{-}^{p_{\lambda }^{\left(s\right)}},m_{+}^{p_{\lambda }^{\left(s\right)}}\right)\right\}.$$

Now, we state and prove the main result of this section.

Theorem 4. 

For $N\ge 1$, $s>0$ and $\lambda \in (0,N)$, consider ${\mu }_N=\left(1-{\displaystyle \frac{\lambda }{N}}\right){\delta }_0+{\displaystyle \frac{\lambda }{N}}{\delta }_s$. Then there is $\epsilon >0$, so that for all $m\in (s\lambda -\epsilon ,s\lambda +\epsilon )$

$${\mathbb{Q}}_m^{{\mu }_N^{⊞N}}\stackrel{N\to +\infty }{\to }{\mathbb{Q}}_m^{p_{\lambda }^{\left(s\right)}},in distribution.$$

For $m=s\lambda$, we obtain the FP limit theorem, that is ${\mu }_N^{⊞N}\stackrel{N\to +\infty }{\to }{p}_{\lambda }^{\left(s\right)},$ in distribution.

Proof. 

One has $m_1^{{\mu }_N^{⊞N}}=s\lambda =m_1^{p_{\lambda }^{\left(s\right)}}$. Then, there exists $\epsilon >0$ such that

$$\left(m_{-}^{{\mu }_N^{⊞N}},m_{+}^{{\mu }_N^{⊞N}}\right)\cap \left(m_{-}^{p_{\lambda }^{\left(s\right)}},m_{+}^{p_{\lambda }^{\left(s\right)}}\right)=(s\lambda -\epsilon ,s\lambda +\epsilon ).$$

For all $m\in (s\lambda -\epsilon ,s\lambda +\epsilon )$, we have

$$\int t{Q}_m^{{\mu }_N^{⊞N}}\left(dt\right)=m=\int t{Q}_m^{p_{\lambda }^{\left(s\right)}}\left(dt\right).$$

Using the VFs and the fact that ${\mathcal{V}}_{{\mu }_N}\left(m\right)=m(s-m)$ (see [23] (p. 878)), ∀$m\in (s\lambda -\epsilon ,s\lambda +\epsilon )$, we have

$${\mathfrak{V}}_{{\mu }_N^{⊞N}}\left(m\right)=N{\mathfrak{V}}_{{\mu }_N}(m/N)=m(s-m/N)\stackrel{N\to +\infty }{\to }sm={\mathfrak{V}}_{p_{\lambda }^{\left(s\right)}}\left(m\right).$$

This along with [13] (Proposition 4.2) applied to measures $Q_m^{{\mu }_N^{⊞N}}$ gives

$${\mathbb{Q}}_m^{{\mu }_N^{⊞N}}\stackrel{N\to +\infty }{\to }{\mathbb{Q}}_m^{p_{\lambda }^{\left(s\right)}},\mathrm{in}\mathrm{distribution},\forall m\in (s\lambda -\epsilon ,s\lambda +\epsilon ).$$

For $m=s\lambda$, we have the FP limit theorem, which is ${\mu }_N^{⊞N}\stackrel{N\to +\infty }{\to }{p}_{\lambda }^{\left(s\right)},$ in distribution. □

5. Limit Theorems Related to Fermi Convolution

Studying limit theorems related to Fermi convolution is essential for understanding the asymptotic behavior of non-commutative random variables governed by fermionic statistics. Fermi convolution arises in the context of free probability and quantum probability, where classical notions of independence are replaced by anti-commutation relations. Limit theorems in this setting provide deep insight into the distributional behavior of large systems of fermionic particles, with applications in quantum physics, statistical mechanics, and operator algebras. They help generalize classical probabilistic results to non-commutative frameworks, offering a powerful analytical tool to model and analyze complex quantum systems. In essence, these theorems bridge the gap between abstract mathematical structures and physical phenomena, making them a vital area of study in both pure and applied mathematics.

To understand the conclusions of this section, we need to review some fundamental principles regarding Fermi convolution mentioned in [24]. ${\mathcal{P}}^2$ and ${\mathcal{P}}_c$ are subsets of probability measures from $\mathcal{P}$ with finite mean and variance, and compact support, respectively. The $\tilde{B}$-transform is defined in [24] for $\varrho \in {\mathcal{P}}^2$, as

$${\tilde{B}}_{\varrho }\left(w\right)=m_1^{\varrho }w+1-{\displaystyle \frac{w}{{\mathbb{G}}_{{\varrho }^0}\left(1/w\right)}}.$$

(48)

where measure ${\varrho }^0$ represents the zero mean shift of measure $\varrho$.

Let $\varrho ={\varrho }_1•{\varrho }_2$ be the Fermi convolution of ${\varrho }_1$ and ${\varrho }_2\in {\mathcal{P}}^2$. From [24] (Theorem 3.1) we have

$${\tilde{B}}_{\varrho }\left(w\right)={\tilde{B}}_{{\varrho }_1}\left(w\right)+{\tilde{B}}_{{\varrho }_2}\left(w\right).$$

Furthermore, $\varrho \in {\mathcal{P}}^2$ and $m_1^{\varrho }=m_1^{{\varrho }_1}+m_1^{{\varrho }_2}$.

We say that $\varrho \in {\mathcal{P}}^2$ is •-infinitely divisible if for each $p\in \mathbb{N}$, there is ${\varrho }_p\in {\mathcal{P}}^2$ so that

$$\varrho =\underset{p\mathrm{times}}{\underbrace{{\varrho }_p•.....•{\varrho }_p}}.$$

All probabilities in ${\mathcal{P}}^2$ are •-infinitely divisible; see [24] (Remark 3.2).

The following finding is important for proving Theorem 5.

Proposition 1. 

Suppose $\varrho \in {\mathcal{P}}^2$ with $sup\mathrm{supp}\left(\varrho \right)<+\infty$. For $\alpha >0$ such that ${\left({\varrho }^{⊞1/\alpha }\right)}^{•\alpha }$ and ${\left({\varrho }^{•1/\alpha }\right)}^{⊞\alpha }$ are defined and for $m>m_1^{\varrho }$ close sufficiently to $m_1^{\varrho }$, one has

$${\mathbb{V}}_{{\left({\varrho }^{⊞1/\alpha }\right)}^{•\alpha }}\left(m\right)=(1/\alpha -1)m(m-m_1^{\varrho })+{\mathbb{V}}_{\varrho }\left(m\right)$$

(49)

and

$${\mathbb{V}}_{{\left({\varrho }^{•1/\alpha }\right)}^{⊞\alpha }}\left(m\right)=(1-1/\alpha )m(m-m_1^{\varrho })+{\mathbb{V}}_{\varrho }\left(m\right).$$

(50)

Furthermore,

$${\mathfrak{V}}_{{\left({\varrho }^{⊞1/\alpha }\right)}^{•\alpha }}\left(m\right)=(1/\alpha -1){(m-m_1^{\varrho })}^2+{\mathfrak{V}}_{\varrho }\left(m\right)$$

(51)

and

$${\mathfrak{V}}_{{\left({\varrho }^{•1/\alpha }\right)}^{⊞\alpha }}\left(m\right)=(1-1/\alpha ){(m-m_1^{\varrho })}^2+{\mathfrak{V}}_{\varrho }\left(m\right).$$

(52)

Proof. 

If $\varrho \in {\mathcal{P}}_{ba}$, then ${\varrho }^{⊞1/\alpha }\in {\mathcal{P}}_{ba}$ (see [14] (Proposition 3.10) or [15] (Proposition 6.1)). This together with (48) implies that ${\left({\varrho }^{⊞1/\alpha }\right)}^{•\alpha }\in {\mathcal{P}}_{ba}$. In addition, basing on [25] (Equation (18)) and (7), for m close to $m_1^{{\left({\varrho }^{⊞1/\alpha }\right)}^{•\alpha }}=m_1^{\rho }$, we have

$$\begin{array}{ccc}\hfill {\mathbb{V}}_{{\left({\varrho }^{⊞1/\alpha }\right)}^{•\alpha }}\left(m\right)& =& \alpha {\mathbb{V}}_{{\varrho }^{⊞1/\alpha }}(m/\alpha )+(1/\alpha -1)m^2+m_1^{\left({\varrho }^{⊞1/\alpha }\right)}(\alpha -1)m\hfill \\ & =& {\mathbb{V}}_{\varrho }\left(m\right)+(1/\alpha -1)m^2+m_1^{\varrho }(1-1/\alpha )m,\hfill \end{array}$$

which is nothing but (49). Furthermore, relation (51) follows from (1) and (49). The same arguments are used for ${\left({\varrho }^{•1/\alpha }\right)}^{⊞\alpha }$ to obtain formulas (50) and (52). □

Corollary 1. 

Suppose $\varrho \in {\mathcal{P}}^2$ with $sup\mathrm{supp}\left(\varrho \right)<+\infty$. For $t\ge 0$, the measure ${\mathbb{A}}_t\left(\varrho \right)={\left({\varrho }^{⊞(1+t)}\right)}^{•{\displaystyle \frac{1}{1+t}}}\in {\mathcal{P}}_{ba}$ and, for m close to $m_1^{{\mathbb{A}}_t\left(\varrho \right)}=m_1^{\rho }$, we have

$${\mathbb{V}}_{{\mathbb{A}}_t\left(\varrho \right)}\left(m\right)=tm(m-m_1^{\varrho })+{\mathbb{V}}_{\varrho }\left(m\right),$$

(53)

and

$${\mathfrak{V}}_{{\mathbb{A}}_t\left(\varrho \right)}\left(m\right)=t{(m-m_1^{\varrho })}^2+{\mathfrak{V}}_{\varrho }\left(m\right).$$

(54)

Proof. 

The proof follows from (49) and (51) by taking $\alpha =1/(1+t).$ □

Denote by $\mathbf{V}$ the class of VFs that correspond to $\varrho \in {\mathcal{P}}_c$. Denote by ${\mathbf{V}}_{•}$ the class of those ${\mathfrak{V}}_{\varrho }\in \mathbf{V}$ such that the corresponding probability measures $\varrho$ are •-infinitely divisible. Since all real probability measures are •-infinitely divisible, then ${\mathbf{V}}_{•}=\mathbf{V}$. Denote by ${\mathbf{V}}_{⊞}$ the class of those ${\mathfrak{V}}_{\varrho }\in \mathbf{V}$ such that the corresponding probability measures $\varrho$ are ⊞-infinitely divisible. We have the following result.

Theorem 5. 

Suppose $\varrho \in {\mathcal{P}}_c$. For $\alpha >0$ such that ${\left({\varrho }^{⊞1/\alpha }\right)}^{•\alpha }$ and ${\left({\varrho }^{•1/\alpha }\right)}^{⊞\alpha }$ are defined, we have

(i) 

The map ${\mathfrak{V}}_{\varrho }\left(m\right)\mapsto {\mathfrak{V}}_{\varrho }\left(m\right)+{(m-m_1^{\varrho })}^2$ is a bijection from $\mathbf{V}$ onto ${\mathbf{V}}_{⊞}$ and

$${\left({\varrho }^{•1/\alpha }\right)}^{⊞\alpha }\stackrel{\alpha \to +\infty }{\to }{\mathbb{A}}_1\left(\varrho \right),in distribution.$$

(55)

(ii) 

The map ${\mathfrak{V}}_{\varrho }\left(m\right)\mapsto {\mathfrak{V}}_{\varrho }\left(m\right)-{(m-m_1^{\varrho })}^2$ is a bijection from ${\mathbf{V}}_{⊞}$ onto $\mathbf{V}$ and

$${\left({\varrho }^{⊞1/\alpha }\right)}^{•\alpha }\stackrel{\alpha \to +\infty }{\to }{\mathbb{A}}_1^{-1}\left(\varrho \right),in distribution.$$

(56)

Proof. 

(i) It is clear from (54) that the map ${\mathfrak{V}}_{\varrho }\left(m\right)\mapsto {\mathfrak{V}}_{\varrho }\left(m\right)+{(m-m_1^{\varrho })}^2$ correspond to the bijection $\varrho \mapsto {\mathbb{A}}_1\left(\varrho \right)={\left({\varrho }^{⊞2}\right)}^{•{\displaystyle \frac{1}{2}}}$. In other words, if ${\mathfrak{V}}_{\varrho }(·)$ is the VF of the CSK family generated by $\varrho \in {\mathcal{P}}_c$ which is •-infinitely divisible, then ${\mathfrak{V}}_{\varrho }\left(m\right)+{(m-m_1^{\varrho })}^2$ is the VF of the CSK family generated by ${\mathbb{A}}_1\left(\varrho \right)$ which is ⊞-infinitely divisible. So that, the map ${\mathfrak{V}}_{\varrho }\left(m\right)\mapsto {\mathfrak{V}}_{\varrho }\left(m\right)+{(m-m_1^{\varrho })}^2$ is a bijection from $\mathbf{V}$ onto ${\mathbf{V}}_{⊞}$. Furthermore, one sees from (52) that

$${\displaystyle \underset{\alpha \to +\infty }{lim}{\mathfrak{V}}_{{\left({\varrho }^{•1/\alpha }\right)}^{⊞\alpha }}\left(m\right)={\mathfrak{V}}_{\varrho }\left(m\right)+{(m-m_1^{\varrho })}^2={\mathfrak{V}}_{{\mathbb{A}}_1\left(\varrho \right)}\left(m\right),}$$

which implies (55), by the use of [13] (Proposition 4.2).

(ii) The map ${\mathfrak{V}}_{\varrho }\left(m\right)\mapsto {\mathfrak{V}}_{\varrho }\left(m\right)-{(m-m_1^{\varrho })}^2$ correspond to the bijection $\varrho ⟼{\mathbb{A}}_1^{-1}\left(\varrho \right)={\left({\varrho }^{•2}\right)}^{⊞{\displaystyle \frac{1}{2}}}$. In addition, one sees from (51) that

$${\displaystyle \underset{\alpha \to +\infty }{lim}{\mathfrak{V}}_{{\left({\varrho }^{⊞1/\alpha }\right)}^{•\alpha }}\left(m\right)={\mathfrak{V}}_{\varrho }\left(m\right)-{(m-m_1^{\varrho })}^2={\mathfrak{V}}_{{\mathbb{A}}_1^{-1}\left(\varrho \right)}\left(m\right),}$$

which implies (56), by the use of [13] (Proposition 4.2). □

The following result incorporates Boolean and Fermi convolutions and is useful in the proof of Theorem 6.

Proposition 2. 

Suppose $\varrho \in {\mathcal{P}}^2$ with $sup\mathrm{supp}\left(\varrho \right)<+\infty$. For $m>m_1^{\varrho }$ close enough to $m_1^{\varrho }$

$${\mathbb{V}}_{{\left({\varrho }^{\uplus 1/\alpha }\right)}^{•\alpha }}\left(m\right)={\mathbb{V}}_{\varrho }\left(m\right)+m_1^{\varrho }(1-1/\alpha )m$$

(57)

and

$${\mathbb{V}}_{{\left({\varrho }^{•1/\alpha }\right)}^{\uplus \alpha }}\left(m\right)={\mathbb{V}}_{\varrho }\left(m\right)+m_1^{\varrho }(1/\alpha -1)m.$$

(58)

Furthermore, we have

$${\mathfrak{V}}_{{\left({\varrho }^{\uplus 1/\alpha }\right)}^{•\alpha }}\left(m\right)={\mathfrak{V}}_{\varrho }\left(m\right)+m_1^{\varrho }(1-1/\alpha )(m-m_1^{\varrho })$$

(59)

and

$${\mathfrak{V}}_{{\left({\varrho }^{•1/\alpha }\right)}^{\uplus \alpha }}\left(m\right)={\mathfrak{V}}_{\varrho }\left(m\right)+m_1^{\varrho }(1/\alpha -1)(m-m_1^{\varrho }).$$

(60)

Proof. 

If $\varrho \in {\mathcal{P}}_{ba}$, then ${\varrho }^{\uplus 1/\alpha }\in {\mathcal{P}}_{ba}$ (see [18] (Theorem 3.2 (i))). This together with (48) implies that ${\left({\varrho }^{\uplus 1/\alpha }\right)}^{•\alpha }\in {\mathcal{P}}_{ba}$. In addition, basing on [25] (Equation (18)) and (9), for m close to $m_1^{{\left({\varrho }^{\uplus 1/\alpha }\right)}^{•\alpha }}=m_1^{\rho }$, we have

$$\begin{array}{ccc}\hfill {\mathbb{V}}_{{\left({\varrho }^{\uplus 1/\alpha }\right)}^{•\alpha }}\left(m\right)& =& \alpha {\mathbb{V}}_{{\varrho }^{\uplus 1/\alpha }}(m/\alpha )+(1/\alpha -1)m^2+m_1^{{\varrho }^{\uplus 1/\alpha }}(\alpha -1)m\hfill \\ & =& {\mathbb{V}}_{\varrho }\left(m\right)+(1-1/\alpha )m^2+(1/\alpha -1)m^2+m_1^{\varrho }(1-1/\alpha )m\hfill \\ & =& {\mathbb{V}}_{\varrho }\left(m\right)+m_1^{\varrho }(1-1/\alpha )m,\hfill \end{array}$$

which is nothing but (57). Furthermore, relation (59) follows from (1) and (57). The same arguments are used for ${\left({\varrho }^{•1/\alpha }\right)}^{\uplus \alpha }$ to obtain formulas (58) and (60). □

Corollary 2. 

Suppose $\varrho \in {\mathcal{P}}^2$ with $sup\mathrm{supp}\left(\varrho \right)<+\infty$. For $t\ge 0$, ${\mathbb{C}}_t\left(\varrho \right)={\left({\varrho }^{\uplus (1+t)}\right)}^{•{\displaystyle \frac{1}{1+t}}}\in {\mathcal{P}}_{ba}$ and for $m>m_1^{\varrho }$ close sufficiently to $m_1^{\varrho }$

$${\mathbb{V}}_{{\mathbb{C}}_t\left(\varrho \right)}\left(m\right)={\mathbb{V}}_{\varrho }\left(m\right)-t{m}_1^{\varrho }m,$$

(61)

and

$${\mathfrak{V}}_{{\mathbb{C}}_t\left(\varrho \right)}\left(m\right)={\mathfrak{V}}_{\varrho }\left(m\right)-t{m}_1^{\varrho }(m-m_1^{\varrho }).$$

(62)

Proof. 

The proof follows from (57) and (59) by taking $\alpha =1/(1+t).$ □

Denote by ${\mathbf{V}}_{\uplus }$ the class of those ${\mathfrak{V}}_{\varrho }\in \mathbf{V}$ such that the corresponding probability measures $\varrho$ are ⊎-infinitely divisible. Since all real probability measures are ⊎-infinitely divisible, then ${\mathbf{V}}_{\uplus }=\mathbf{V}$.

Theorem 6. 

Suppose $\varrho \in {\mathcal{P}}_c$. We have

(i) 

The map ${\mathfrak{V}}_{\varrho }\left(m\right)\mapsto {\mathfrak{V}}_{\varrho }\left(m\right)-m_1^{\varrho }(m-m_1^{\varrho })$ is a bijection from $\mathbf{V}$ onto $\mathbf{V}$ and

$${\left({\varrho }^{•1/\alpha }\right)}^{\uplus \alpha }\stackrel{\alpha \to +\infty }{\to }{\mathbb{C}}_1\left(\varrho \right),in distribution.$$

(63)

(ii) 

The map ${\mathfrak{V}}_{\varrho }\left(m\right)\mapsto {\mathfrak{V}}_{\varrho }\left(m\right)+m_1^{\varrho }(m-m_1^{\varrho })$ is a bijection from $\mathbf{V}$ onto $\mathbf{V}$ and

$${\left({\varrho }^{\uplus 1/\alpha }\right)}^{•\alpha }\stackrel{\alpha \to +\infty }{\to }{\mathbb{C}}_1^{-1}\left(\varrho \right),in distribution.$$

(64)

Proof. 

(i) It is clear from (62) that the map ${\mathfrak{V}}_{\varrho }\left(m\right)\mapsto {\mathfrak{V}}_{\varrho }\left(m\right)-m_1^{\varrho }(m-m_1^{\varrho })$ corresponds to the bijection $\varrho \mapsto {\mathbb{C}}_1\left(\varrho \right)={\left({\varrho }^{\uplus 2}\right)}^{•{\displaystyle \frac{1}{2}}}$. In other words, if ${\mathfrak{V}}_{\varrho }\left(m\right)$ is the VF of the CSK family generated by $\varrho \in {\mathcal{P}}_c$ which is •-infinitely divisible, then ${\mathfrak{V}}_{\varrho }\left(m\right)-m_1^{\varrho }(m-m_1^{\varrho })$ is the VF of the CSK family generated by ${\mathbb{C}}_1\left(\varrho \right)$ which is ⊎-infinitely divisible. Thus, the map ${\mathfrak{V}}_{\varrho }\left(m\right)\mapsto {\mathfrak{V}}_{\varrho }\left(m\right)-m_1^{\varrho }(m-m_1^{\varrho })$ is a bijection from $\mathbf{V}$ onto $\mathbf{V}$. In addition, one sees from (60) that

$${\displaystyle \underset{\alpha \to +\infty }{lim}{\mathfrak{V}}_{{\left({\varrho }^{•1/\alpha }\right)}^{\uplus \alpha }}\left(m\right)={\mathfrak{V}}_{\varrho }\left(m\right)-m_1^{\varrho }(m-m_1^{\varrho })={\mathfrak{V}}_{{\mathbb{C}}_1\left(\varrho \right)}\left(m\right),}$$

which implies (63), by the use of [13] (Proposition 4.2).

(ii) The map ${\mathfrak{V}}_{\varrho }\left(m\right)\mapsto {\mathfrak{V}}_{\varrho }\left(m\right)+m_1^{\varrho }(m-m_1^{\varrho })$ correspond to the bijection $\varrho \mapsto {\mathbb{C}}_1^{-1}\left(\varrho \right)={\left({\nu }^{•2}\right)}^{\uplus {\displaystyle \frac{1}{2}}}$. In addition, one sees from (59) that

$${\displaystyle \underset{\alpha \to +\infty }{lim}{\mathfrak{V}}_{{\left({\varrho }^{\uplus 1/\alpha }\right)}^{•\alpha }}\left(m\right)={\mathfrak{V}}_{\varrho }\left(m\right)+m_1^{\varrho }(m-m_1^{\varrho })={\mathfrak{V}}_{{\mathbb{C}}_1^{-1}\left(\varrho \right)}\left(m\right),}$$

which implies (64), by the use of [13] (Proposition 4.2). □

Next, we highlight the importance of Theorems 5 and 6 by considering some specific measures.

Example 1. 

Let

$$s{c}_{\eta ,\gamma }\left(dx\right)={\displaystyle \frac{\sqrt{4{\gamma }^2-{(x-\eta )}^2}}{2\pi {\gamma }^2}}{1}_{|x-\eta |<2\gamma }\left(dx\right)$$

denote the semi-circle (SC) law of mean η and variance ${\gamma }^2$. We know that ${\mathfrak{V}}_{s{c}_{\eta ,\gamma }}(·)={\gamma }^2$. For simplicity and without loss of generality, we may suppose that $\gamma =1$ and $\eta \ne 0$.

(i) 

We have

$${\left({\left(s{c}_{\eta ,1}\right)}^{•1/\alpha }\right)}^{⊞\alpha }\stackrel{\alpha \to +\infty }{\to }{\mathit{T}}_{\eta }\left(f{h}_{(0,1)}\right),in distribution,$$

(65)

where $f{h}_{(a,b)}$ is the the free analog of hyperbolic type law with parameters $b>0$ and $a^2<4b$ as presented in [13] (Theorem 3.2). That is,

$$f{h}_{(a,b)}\left(dt\right)={\displaystyle \frac{\sqrt{4(1+b)-{(t-a)}^2}}{2\pi (b{t}^2+at+1)}}{1}_{(a-2\sqrt{1+b},a+2\sqrt{1+b})}\left(t\right)dt$$

(66)

with $m_1^{f{h}_{(a,b)}}=0$ and corresponding VF ${\mathfrak{V}}_{f{h}_{(a,b)}}\left(m\right)=1+am+b{m}^2$, where $b>0$ and $a^2<4b$.

Relation (65) can be seen by means of the VFs. Indeed, for m close to $m_1^{{\left({\left(s{c}_{\eta ,1}\right)}^{•1/\alpha }\right)}^{⊞\alpha }}=m_1^{s{c}_{\eta ,1}}=\eta$, one has

$${\mathfrak{V}}_{{\left({\left(s{c}_{\eta ,1}\right)}^{•1/\alpha }\right)}^{⊞\alpha }}\left(m\right)\stackrel{t\to +\infty }{\to }{\mathfrak{V}}_{{\mathbb{A}}_1\left(s{c}_{\eta ,1}\right)}\left(m\right)=1+{(m-\eta )}^2={\mathfrak{V}}_{{\mathit{T}}_{\eta }\left(f{h}_{(0,1)}\right)}\left(m\right).$$

(ii) 

We have

$${\left({\left(s{c}_{\eta ,1}\right)}^{⊞1/\alpha }\right)}^{•\alpha }\stackrel{\alpha \to +\infty }{\to }{\mathit{T}}_{\eta }\left(f{b}_{(0,-1)}\right),in distribution,$$

(67)

where $f{b}_{(a,b)}$ is the FB law as presented by (36). Indeed, for m close to $m_1^{{\left({\left(s{c}_{\eta ,1}\right)}^{⊞1/\alpha }\right)}^{•\alpha }}=m_1^{s{c}_{\eta ,1}}=\eta$, one has

$${\mathfrak{V}}_{{\left({\left(s{c}_{\eta ,1}\right)}^{⊞1/\alpha }\right)}^{•\alpha }}\left(m\right)\stackrel{t\to +\infty }{\to }{\mathfrak{V}}_{{\mathbb{A}}_1^{-1}\left(s{c}_{\eta ,1}\right)}\left(m\right)=1-{(m-\eta )}^2={\mathfrak{V}}_{{\mathit{T}}_{\eta }\left(f{b}_{(0,-1)}\right)}\left(m\right).$$

Note that $f{b}_{(0,-1)}$ is reduced to the symmetric Bernoulli measure $sb\left(dx\right)={\displaystyle \frac{1}{2}}{\delta }_{-1}+{\displaystyle \frac{1}{2}}{\delta }_1$.

(iii) 

We have

$${\left({\left(s{c}_{\eta ,1}\right)}^{•1/\alpha }\right)}^{\uplus \alpha }\stackrel{\alpha \to +\infty }{\to }{\mathit{T}}_{\eta }\left(m{p}_{-\eta }\right),in distribution,$$

(68)

where $m{p}_a$ is the Marchenko–Pastur (or FP) type law with $a\ne 0$ as presented in [13] (Theorem 3.2). That is

$$m{p}_a\left(dt\right)={\displaystyle \frac{\sqrt{4-{(t-a)}^2}}{2\pi (at+1)}}{1}_{(a-2,a+2)}\left(t\right)dt+{(1-1/a^2)}^{+}{\delta }_{-1/a}\left(dt\right)$$

(69)

$m_1^{m{p}_a}=0$ and corresponding VF ${\mathfrak{V}}_{m{p}_a}\left(m\right)=1+am$, where $a\ne 0$. Note that the FP law $p_{\lambda }^{\left(s\right)}$ provided by (45) is just an affine transformation of $m{p}_a$ (of the form (69) with $a=\pm 1/\sqrt{\lambda }$) by the map $t\mapsto s\lambda (\pm {\displaystyle \frac{1}{\sqrt{\lambda }}}t+1)$.

Relation (68) can be seen by means of the VFs. Indeed, for m close to $m_1^{{\left({\left(s{c}_{\eta ,1}\right)}^{•1/\alpha }\right)}^{\uplus \alpha }}=m_1^{s{c}_{\eta ,1}}=\eta$, one has

$${\mathfrak{V}}_{{\left({\left(s{c}_{\eta ,1}\right)}^{•1/\alpha }\right)}^{\uplus \alpha }}\left(m\right)\stackrel{t\to +\infty }{\to }{\mathfrak{V}}_{{\mathbb{C}}_1\left(s{c}_{\eta ,1}\right)}\left(m\right)=1-\eta (m-\eta )={\mathfrak{V}}_{{\mathit{T}}_{\eta }\left(m{p}_{-\eta }\right)}\left(m\right).$$

(iv) 

We have

$${\left({\left(s{c}_{\eta ,1}\right)}^{\uplus 1/\alpha }\right)}^{•\alpha }\stackrel{\alpha \to +\infty }{\to }{\mathit{T}}_{\eta }\left(m{p}_{\eta }\right),in distribution.$$

Indeed, for m close to $m_1^{{\left({\left(s{c}_{\eta ,1}\right)}^{\uplus 1/\alpha }\right)}^{•\alpha }}=m_1^{s{c}_{\eta ,1}}=\eta$, one has

$${\mathfrak{V}}_{{\left({\left(s{c}_{\eta ,1}\right)}^{\uplus 1/\alpha }\right)}^{•\alpha }}\left(m\right)\stackrel{t\to +\infty }{\to }{\mathfrak{V}}_{{\mathbb{C}}_1^{-1}\left(s{c}_{\eta ,1}\right)}\left(m\right)=1+\eta (m-\eta )={\mathfrak{V}}_{{\mathit{T}}_{\eta }\left(m{p}_{\eta }\right)}\left(m\right).$$

Example 2. 

The Bernoulli law $ber\left(dx\right)={\displaystyle \frac{1}{2}}{\delta }_0+{\displaystyle \frac{1}{2}}{\delta }_1$ generates the CSK family with $m_1^{ber}={\displaystyle \frac{1}{2}}$ and VF ${\mathfrak{V}}_{ber}\left(m\right)=1-{(m-1/2)}^2$.

(i) 

We have

$${\left({\left(ber\right)}^{•1/\alpha }\right)}^{⊞\alpha }\stackrel{\alpha \to +\infty }{\to }s{c}_{{\displaystyle \frac{1}{2}},1},in distribution.$$

In fact, for m close to $m_1^{{\left({\left(ber\right)}^{•1/\alpha }\right)}^{⊞\alpha }}=m_1^{ber}=1/2$, one has

$${\mathfrak{V}}_{{\left({\left(ber\right)}^{•1/\alpha }\right)}^{⊞\alpha }}\left(m\right)\stackrel{t\to +\infty }{\to }{\mathfrak{V}}_{{\mathbb{A}}_1\left(ber\right)}\left(m\right)=1-{(m-1/2)}^2+{(m-1/2)}^2=1={\mathfrak{V}}_{s{c}_{{\displaystyle \frac{1}{2}},1}}\left(m\right).$$

(ii) 

We have

$${\left({\left(ber\right)}^{•1/\alpha }\right)}^{\uplus \alpha }\stackrel{\alpha \to +\infty }{\to }{\mathit{T}}_{{\displaystyle \frac{1}{2}}}\left(f{b}_{\left(-1/2,-1\right)}\right),in distribution.$$

In fact, for m close to $m_1^{{\left({\left(ber\right)}^{•1/\alpha }\right)}^{\uplus \alpha }}=m_1^{ber}=1/2$, one has

$${\mathfrak{V}}_{{\left({\left(ber\right)}^{•1/\alpha }\right)}^{\uplus \alpha }}\left(m\right)\stackrel{t\to +\infty }{\to }{\mathfrak{V}}_{{\mathbb{C}}_1\left(ber\right)}\left(m\right)=1-{(m-1/2)}^2-{\displaystyle \frac{1}{2}}(m-1/2)={\mathfrak{V}}_{{\mathit{T}}_{{\displaystyle \frac{1}{2}}}\left(f{b}_{\left(-1/2,-1\right)}\right)}\left(m\right).$$

(iii) 

We have

$${\left({\left(ber\right)}^{\uplus 1/\alpha }\right)}^{•\alpha }\stackrel{\alpha \to +\infty }{\to }{\mathit{T}}_{{\displaystyle \frac{1}{2}}}\left(f{b}_{\left(1/2,-1\right)}\right),in distribution.$$

In fact, for m close to $m_1^{{\left({\left(ber\right)}^{\uplus 1/\alpha }\right)}^{•\alpha }}=m_1^{ber}=1/2$, one has

$${\mathfrak{V}}_{{\left({\left(ber\right)}^{\uplus 1/\alpha }\right)}^{•\alpha }}\left(m\right)\stackrel{t\to +\infty }{\to }{\mathfrak{V}}_{{\mathbb{C}}_1^{-1}\left(ber\right)}\left(m\right)=1-{(m-1/2)}^2+{\displaystyle \frac{1}{2}}(m-1/2)={\mathfrak{V}}_{{\mathit{T}}_{{\displaystyle \frac{1}{2}}}\left(f{b}_{\left(1/2,-1\right)}\right)}\left(m\right).$$

6. Conclusions

The CSK families of probability measures, defined through the analytic properties of the Cauchy–Stieltjes transform, provide a rich and unifying framework for exploring a wide range of probabilistic phenomena. These families generalize exponential families in non-classical settings and have found deep connections to areas such as free probability, random matrix theory, and complex analysis. A key component in understanding the structure and behavior of such families is the relative VFs, which encapsulates the interplay between the underlying measure and its associated transform. This function not only governs local and global fluctuations but also offers insight into the geometry of the parameter space and the stability of statistical inference methods.

In this work, we have carried out a detailed investigation into several theoretical properties of CSK families. Our analysis highlights their fundamental role in the framework of modern probability theory, while also underlining their potential to stimulate progress in both theoretical developments and applied research. A central part of our contribution lies in establishing new properties of the FG law, which we derived by exploiting the interplay between free and Boolean additive convolutions. Alongside this, we examined the FB CSK family, where we identified a specific property linked to location parameter. Importantly, we also demonstrated that there cannot exist a CSK family characterized by a scale parameter, which provides new insights into the structural limitations of these families. Moreover, we proposed an estimation method for elements of the FP CSK family, relying on techniques from free additive convolution. Building on this foundation, we employed VFs to establish new limit theorems related to the Fermi convolution, incorporating tools from both free and Boolean probability. These new limit theorems draw a coherent and structured connection between several cornerstone distributions, including the SC, FP, and FB laws. This pathway not only enriches the understanding of CSK families but also reinforces their central position in the broader landscape of free probability theory.

In summary, the study of CSK families of probability measures is critical to developing the theoretical and computational framework of free harmonic analysis. These families contain detailed information about the behavior and properties of noncommutative probability distributions. Their structural qualities make detailed modeling of free convolutions possible, as well as the creation of analytic tools for understanding spectral distributions in random matrix theory and operator algebra. Furthermore, by investigating the related VFs and functional transformations, researchers can develop reliable methods for inference, estimation, and asymptotic analysis in free probabilistic systems. As such, further investigation of these families is critical for connecting abstract free probability theory to real applications in mathematics, physics, and statistics.