Notes on the Free Additive Convolution

Abstract

The investigation of free additive convolution is a key concept in free probability theory, offering a framework for studying the sum of freely independent random variables. This paper uses free additive convolution and measure dilations to investigate various aspects of Marchenko–Pastur and free Gamma laws in the setting of Cauchy-Stieltjes Kernel (CSK) families. Our investigation reveals the essential links between analytic function theory and free probability, highlighting the usefulness of CSK families in developing the theoretical and computational aspects of free additive convolution.

1. Introduction

Free additive convolution ⊞ is a concept from free probability theory, which extends classical probability to non-commutative settings, particularly in the study of random matrices and operator algebras. It describes the addition of freely independent random variables, analogous to classical convolution for independent variables in probability theory. Free additive convolution has significant applications in random matrix theory, particularly in studying the asymptotic eigenvalue distributions of large random matrices, see [1,2,3]. The aim of this article is to advance the understanding of the free additive convolution within the context of Cauchy-Stieltjes Kernel (CSK) families and the associated variance functions (VFs). We discuss some mathematical formulations of free additive convolution and its computational aspects. We explore the interplay between free additive convolution and CSK families, focusing on the analytic representations, computational techniques, and structural properties of the resulting distributions. To better understand the objective of this paper, we must first introduce some fundamental principles concerning free additive convolution [4] and CSK families. We denote $\mathcal{P}$ as the set of non-degenerate probability measures on $\mathbb{R}$. The measure $\tau ⊞\varrho$ represents the additive free convolution of $\tau$ and $\varrho \in \mathcal{P}$, and it verifies

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

where the free cumulant transform, ${\mathfrak{R}}_{\tau }$, of $\tau$ is defined as

$${\mathfrak{R}}_{\tau }\left({\mathbb{G}}_{\tau }\left(w\right)\right)=w-1/{\mathbb{G}}_{\tau }\left(w\right),w\mathrm{in} \mathrm{an} \mathrm{appropriate} \mathrm{domain},$$

and

$${\mathbb{G}}_{\tau }\left(w\right)=\int {\displaystyle \frac{\tau \left(d\zeta \right)}{w-\zeta }},w\in \mathbb{C}\setminus \mathrm{supp}\left(\tau \right)$$

is the Cauchy transform of $\tau$.

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

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

The t-fold free additive convolution of $\tau$ with itself is represented as ${\tau }^{⊞t}$. It is well-defined ∀ $t\ge 1$ [5], and

$${\mathfrak{R}}_{{\tau }^{⊞t}}\left(\xi \right)=t{\mathfrak{R}}_{\tau }\left(\xi \right).$$

(1)

A measure $\tau \in \mathcal{P}$ is infinitely divisible with respect to ⊞ if ${\tau }^{⊞t}$ is well-defined ∀$t>0$.

We come now to the concept of CSK families: This concept revolves around families of probability measures defined in a manner similar to that of natural exponential families (NEFs) by incorporating the Cauchy–Stieltjes kernel ${\displaystyle \frac{1}{1-\theta l}}$ in place of the exponential kernel $exp\left(\theta l\right)$. The CSK families have been studied in [6,7] for measures with compact support. Further properties were proved in [8] by involving measures having a one-sided support boundary, for example, from above. The CSK families of probabilities play a crucial role in free probability theory, particularly in the study of noncommutative probability measures. These families, characterized by their VFs, provide a powerful analytical tool for describing the distribution of free convolutions and asymptotic spectral properties of large random matrices. Their utility extends to solving moment problems, deriving limit theorems, and characterizing free infinitely divisible distributions. By leveraging Cauchy–Stieltjes transforms, researchers gain deeper insights into free harmonic analysis, operator algebraic structures, and interesting links to random matrix theory. ${\mathcal{P}}_{ba}$ refers to the subset of $\mathcal{P}$ that includes probability measures with a one-sided support border from above. Let $\rho \in {\mathcal{P}}_{ba}$; then, with $1/{\theta }_{+}^{\rho }=max\{sup\mathrm{supp}\left(\rho \right),0\}$, the function

$${\mathbb{M}}^{\rho }\left(\theta \right)=\int {\displaystyle \frac{\rho \left(dl\right)}{1-\theta l}}$$

is finite for $0\le \theta <{\theta }_{+}^{\rho }$. The CSK family (of probability measures) is the set

$${\mathfrak{F}}_{+}\left(\rho \right)=\left\{{\mathbb{P}}^{(\theta ,\rho )}\left(dl\right)={\displaystyle \frac{\rho \left(dl\right)}{{\mathbb{M}}^{\rho }\left(\theta \right)(1-\theta l)}}:0<\theta <{\theta }_{+}^{\rho }\right\}.$$

The map $\theta \mapsto {\mathbb{K}}^{\rho }\left(\theta \right)=\int l{\mathbb{P}}^{(\theta ,\rho )}\left(dl\right)$ is bijective from $(0,{\theta }_{+}^{\rho })$ into the interval $(m_1^{\rho },m_{+}^{\rho })={\mathbb{K}}^{\rho }\left((0,{\theta }_{+}^{\rho })\right)$, which is the mean domain of ${\mathfrak{F}}_{+}\left(\rho \right)$. The inverse of ${\mathbb{K}}^{\rho }(·)$ is denoted as ${\Phi }^{\rho }(·)$. For $s\in (m_1^{\rho },m_{+}^{\rho })$, we consider ${\mathbb{Q}}_s^{\rho }\left(dl\right)={\mathbb{P}}^{({\Phi }^{\rho }\left(s\right),\rho )}\left(dl\right)$. Then, ${\mathfrak{F}}_{+}\left(\rho \right)=\{{\mathbb{Q}}_s^{\rho }\left(dl\right):m_1^{\rho }<s<m_{+}^{\rho }\}$ represent the mean parametrization. We have $m_1^{\rho }={lim}_{\theta \to 0^{+}}{\mathbb{K}}^{\rho }\left(\theta \right)$ and $m_{+}^{\rho }={\mathbb{B}}_{\rho }-{lim}_{w\to {\mathbb{B}}_{\rho }^{+}}1/{\mathbb{G}}_{\rho }\left(w\right)$ with ${\mathbb{B}}_{\rho }=1/{\theta }_{+}^{\rho }$; see [8].

If a measure $\rho$ has a one-sided support boundary from below, the relative CSK family is denoted as ${\mathfrak{F}}_{-}\left(\rho \right)$, and ${\theta }_{-}^{\rho }<\theta <0$, where ${\theta }_{-}^{\rho }$ is either $1/{\mathbb{A}}_{\rho }$ or $-\infty$ with ${\mathbb{A}}_{\rho }=min\{inf\mathrm{supp}\left(\rho \right),0\}$. The mean domain for ${\mathfrak{F}}_{-}\left(\rho \right)$ is the interval $(m_{-}^{\rho },m_1^{\rho })$ with $m_{-}^{\rho }={\mathbb{A}}_{\rho }-1/{\mathbb{G}}_{\rho }\left({\mathbb{A}}_{\rho }\right)$. If the support of $\rho$ is compact, then $\mathfrak{F}\left(\rho \right)={\mathfrak{F}}_{+}\left(\rho \right)\cup {\mathfrak{F}}_{-}\left(\rho \right)\cup \left\{\rho \right\}$ is the two-sided CSK family.

The VF is given by [7]

$$s\mapsto {\mathfrak{V}}_{\rho }\left(s\right)=\int {(l-s)}^2{\mathbb{Q}}_s^{\rho }\left(dl\right).$$

If $\rho \in {\mathcal{P}}_{ba}$ does not have the first moment, then in ${\mathfrak{F}}_{+}\left(\rho \right)$ all measures have infinite variance. A notion of pseudo-variance function (PVF) ${\mathbb{V}}_{\rho }(·)$ is introduced in [8] as ${\mathbb{V}}_{\rho }\left(s\right)=s\left(1/{\Phi }_{\rho }\left(s\right)-s\right).$ If $m_1^{\rho }=\int l\rho \left(dl\right)$ is finite, then the VF exists [8], and

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

(2)

Now, we present in more detail the objectives of this article: Let ${\mathfrak{F}}_{+}\left(\tau \right)=\{{\mathbb{Q}}_m^{\tau }\left(d\zeta \right);m\in (m_1^{\tau },m_{+}^{\tau })\}$ be the CSK family induced by $\tau \in {\mathcal{P}}_{ba}$ having the finite first moment $m_1^{\tau }$. We denote by $D_c\left(\tau \right)$ the measure dilation of $\tau$ by $c\ne 0$ (i.e., $D_c\left(\tau \right)\left(A\right)=\tau (A/c)$ for $A\subset \mathbb{R}$). For $t>1$, we define the sets of probabilities

$${\left({\mathfrak{F}}_{+}\left(\tau \right)\right)}^{⊞t}=\left\{{\left({\mathbb{Q}}_s^{\tau }\right)}^{⊞t}\left(dl\right):s\in (m_1^{\tau },m_{+}^{\tau })\right\}.$$

(3)

$$D_{1/t}\left({\mathfrak{F}}_{+}\left(\tau \right)\right)=\left\{D_{1/t}\left({\mathbb{Q}}_s^{\tau }\right)\left(dl\right):s\in (m_1^{\tau },m_{+}^{\tau })\right\}.$$

(4)

$$D_{1/t}\left({\left({\mathfrak{F}}_{+}\left(\tau \right)\right)}^{⊞t}\right)=\left\{D_{1/t}\left({\left({\mathbb{Q}}_s^{\tau }\right)}^{⊞t}\right)\left(dl\right):s\in (m_1^{\tau },m_{+}^{\tau })\right\}.$$

(5)

The family ${\left({\mathfrak{F}}_{+}\left(\tau \right)\right)}^{⊞t}$ consists of the t-th free additive convolution powers of each measure ${\mathbb{Q}}_s^{\tau }$ in ${\mathfrak{F}}_{+}\left(\tau \right)$. The question is whether this new family of measures is reduced only (within a dilation) to the CSK family generated by the additive free convolution power of $\tau$. Stated differently: what is the measure $\tau$ satisfying

$${\left({\mathfrak{F}}_{+}\left(\tau \right)\right)}^{⊞t}={\mathfrak{F}}_{+}\left(D_{1/t}\left({\tau }^{⊞t}\right)\right)?$$

(6)

If relation (6) holds true, then $\tau$ is a free Gamma (FG) law up to scaling. Similarly, the family of measures $D_{1/t}\left({\left({\mathfrak{F}}_{+}\left(\tau \right)\right)}^{⊞t}\right)$ is another transformation of the original CSK family, which incorporates free additive convolution and measure dilation. An important issue is to determine the measure $\tau$ for which this transformed family is reduced to the CSK family generated by the additive free convolution power of $\tau$. That is, what is the measure $\tau$ satisfying

$$D_{1/t}\left({\left({\mathfrak{F}}_{+}\left(\tau \right)\right)}^{⊞t}\right)={\mathfrak{F}}_{+}\left({\tau }^{⊞t}\right)?$$

(7)

If relation (7) holds true, then $\tau$ is also an FG law up to scaling. Furthermore, we aim to determine the measure $\tau$ that generates the same CSK family that its measure dilation generates. In other words, what is the measure $\tau$ satisfying

$${\mathfrak{F}}_{+}\left(D_{1/t}\left(\tau \right)\right)={\mathfrak{F}}_{+}\left(\tau \right)?$$

(8)

If relation (8) holds true, then $\tau$ is also an FG law up to scaling. In this context, Theorem 1 will demonstrate how certain conditions, specifically those describing how CSK families behave under free additive convolution and dilation of measures, play a key role in analyzing some features of the Marchenko–Pastur (MP) law. These conditions help identify when a CSK family is stable or transforms in a predictable way under these operations, which is essential for understanding the structural properties of the MP law. Since the MP law is a fundamental object in free probability, exploring its connections to CSK families under convolution and scaling provides insight into its uniqueness and fixed-point behavior. Unlike prior characterizations such as those for the free Meixner families [7], which rely primarily on specific forms of the VF, the results here highlight how stability under convolution and scaling can serve as alternative or complementary criteria. This broader perspective not only extends existing classifications but also deepens the understanding of the role of the MP and FG laws as a fixed point under certain free probabilistic transformations.

The two remarks that follow provide some helpful information to support the major findings of this article.

Remark 1.

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

(i) 

${\mathbb{V}}_{\tau }(·)$ determines τ. Consider $\Delta =\Delta \left(s\right)=s+{\mathbb{V}}_{\rho }\left(s\right)/s$; then,

$${\mathbb{G}}_{\tau }(\Delta )=s/{\mathbb{V}}_{\rho }\left(s\right).$$

(9)

If $m_1^{\tau }$ is finite, then

$${\mathbb{G}}_{\tau }(\Delta )=(s-m_1^{\tau })/{\mathfrak{V}}_{\tau }\left(s\right).$$

(10)

Thus, ${\mathfrak{V}}_{\tau }(·)$ and $m_1^{\tau }$ characterizes τ.

(ii) 

Consider $\chi :\zeta \mapsto \kappa \zeta +\beta$, where $\kappa \ne 0$ and $\beta \in \mathbb{R}$. Then, for s close enough to $m_1^{\chi \left(\tau \right)}=\chi \left(m_1^{\tau }\right)=\kappa m_1^{\tau }+\beta$,

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

(11)

If the VF exists,

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

(12)

Furthermore,

$${\mathfrak{R}}_{\chi \left(\tau \right)}\left(\xi \right)=\kappa {\mathfrak{R}}_{\tau }\left(\kappa \xi \right)+\beta ,\xi close to0.$$

(13)

(iii) 

For $t>0$, so that ${\tau }^{⊞t}$ is defined and for s close enough to $m_1^{{\tau }^{⊞t}}=t{m}_1^{\tau }$, [8]

$${\mathbb{V}}_{{\tau }^{⊞t}}\left(s\right)=t{\mathbb{V}}_{\tau }(s/t).$$

(14)

If the VF exists,

$${\mathfrak{V}}_{{\tau }^{⊞t}}\left(s\right)=t{\mathfrak{V}}_{\tau }(s/t).$$

(15)

Remark 2.

For $\tau \in {\mathcal{P}}_{ba}$, we know that $m_1^{\tau }$ behave nicely under affinity and free additive convolution powers. That is, $m_1^{\chi \left(\tau \right)}=\chi \left(m_1^{\tau }\right)=\kappa m_1^{\tau }+\beta$, and for $t>0$, so that ${\tau }^{⊞t}$ is defined, we have $m_1^{{\tau }^{⊞t}}=t{m}_1^{\tau }$. However, as presented by [9] (Example 3.9) and by [10] (Examples 3.1 and 3.2), there is no general formula for $m_{+}^{\tau }$ when taking the affine transformation or free additive convolution powers of τ. In [9], an extension of the mean domain is presented preserving the PVF (respectively preserving the VF in case of existence). The upper bound of the extended mean domain is introduced as ${\mathit{M}}_{+}^{\tau }=inf\left\{s>m_1^{\tau }:{\mathbb{V}}_{\tau }\left(s\right)/s<0\right\}.$ As explained in [9] (Section 3.2), ${\mathit{M}}_{+}^{\rho }$ behaves nicely under free additive convolution powers: that is, ${\mathit{M}}_{+}^{{\tau }^{⊞t}}=t{\mathit{M}}_{+}^{\tau }$. Note also that, for $\alpha \ne 0$, we have ${\mathit{M}}_{+}^{D_{1/\alpha }\left(\tau \right)}={\mathit{M}}_{+}^{\tau }/\alpha$. In the rest of the paper, for $\tau \in {\mathcal{P}}_{ba}$, we investigate the domain of means $(m_1^{\tau },{\mathit{M}}_{+}^{\tau })$. The extended mean domain is used to guarantee that the parameterizations of the CSK families are well-defined throughout the analysis, especially when the mean parameter approaches boundary values or sits beyond the original family’s natural domain. This expansion provides additional freedom when using operations like convolution and dilation, which might push the mean outside the initial range. Many of the proofs’ arguments involve assessing VFs or at places defined by convolutions or linear transformations of means. The extended domain ensures that these points remain inside an area where the VFs are analytic and their properties are maintained. This assures the validity of functional equations, which are critical to the construction and categorization of kernel families.

2. Notes on the MP Law

For $a\ne 0$, the MP law is provided by

$${\mathbf{mp}}_a\left(dl\right)={\displaystyle \frac{\sqrt{\left({(1+a)}^2-l\right)\left(l-{(a-1)}^2\right)}}{2\pi a^{2}l}}{1}_{\left({(a-1)}^2,{(1+a)}^2\right)}\left(l\right)dl+{\left(1-1/a^2\right)}^{+}{\delta }_0,$$

(16)

with $m_1^{{\mathbf{mp}}_a}=1$. We have ${\mathfrak{V}}_{{\mathbf{mp}}_a}\left(s\right)=a^{2}s.$ The MP law is a fundamental result in random matrix theory and statistical physics, particularly in the study of large-dimensional random matrices; see [11,12]. Named after the mathematicians Vladimir Marchenko and Leonid Pastur, who derived it in the early 1960s, the law describes the limiting spectral distribution (i.e., the distribution of eigenvalues) of some kind of random matrices, especially in the context of large-dimensional systems. The MP law is a key result in the random matrix topic, offering a deep understanding of the limiting behavior of eigenvalues of large random matrices. It has wide applications across many fields, including statistical physics, signal processing, and machine learning, particularly in high-dimensional data analysis. By providing a precise characterization of the spectral distribution, it helps in understanding how random matrices behave as their size grows, providing valuable insights into the structure and properties of large-dimensional systems. On the other hand, the MP law performs the same function in free probability that the Poisson law does in classical probability. It has received much attention from researchers in recent decades. Nonetheless, the literature presents a large number of MP law findings and features: The Lukacs property of the MP law is examined in free probability [13]. In [7], the MP law, as a member of the free Meixner family of probabilities, is distinguished by the matching VF. Furthermore, a brief proof is presented in [14] for the MP theorem. Additional MP law research is provided in [15,16,17]. In this section, based on the ideas of free additive convolution and measure dilation, we present additional features of the MP law in relation to CSK families. To be more specific, we have the following:

Theorem 1.

Let $\tau \in {\mathcal{P}}_{ba}$, with the finite first moment $m_1^{\tau }$. For $t>1$, consider the sets of measures defined by (4) and (5). If

(i) 

$D_{1/t}\left({\mathfrak{F}}_{+}\left(\tau \right)\right)={\mathfrak{F}}_{+}\left(D_{1/t}\left({\tau }^{⊞t}\right)\right)$ or

(ii) 

$D_{1/t}\left({\left({\mathfrak{F}}_{+}\left(\tau \right)\right)}^{⊞t}\right)={\mathfrak{F}}_{+}\left(D_{1/t}\left(\tau \right)\right)$ or

(iii) 

$D_{1/t}\left({\mathfrak{F}}_{+}\left({\tau }^{⊞t}\right)\right)={\mathfrak{F}}_{+}\left(D_{1/t}\left(\tau \right)\right)$ or

(iv) 

${\mathfrak{F}}_{+}\left({\tau }^{⊞t}\right)={\mathfrak{F}}_{+}\left(\tau \right)$,

then τ is a MP law (16) up to scaling.

Proof. 

(i) Assume that $D_{1/t}\left({\mathfrak{F}}_{+}\left(\tau \right)\right)={\mathfrak{F}}_{+}\left(D_{1/t}\left({\tau }^{⊞t}\right)\right)$. Then, for $r\in \left(m_1^{D_{1/t}\left({\tau }^{⊞t}\right)},{\mathbf{M}}_{+}^{D_{1/t}\left({\tau }^{⊞t}\right)}\right)=(m_1^{\tau },{\mathbf{M}}_{+}^{\tau })$, there exists $s\in (m_1^{\tau },{\mathbf{M}}_{+}^{\tau })$, so that

$$D_{1/t}\left({\mathbb{Q}}_s^{\tau }\right)={\mathbb{Q}}_r^{D_{1/t}\left({\tau }^{⊞t}\right)}.$$

(17)

That is,

$${\mathfrak{R}}_{D_{1/t}\left({\mathbb{Q}}_s^{\tau }\right)}\left(\xi \right)={\mathfrak{R}}_{{\mathbb{Q}}_r^{D_{1/t}\left({\tau }^{⊞t}\right)}}\left(\xi \right),\xi \mathrm{close} \mathrm{to}0.$$

(18)

As $m_1^{\tau }$ is finite, the variance of ${\mathbb{Q}}_s^{\tau }$ is finite (i.e $\mathrm{Var}\left({\mathbb{Q}}_s^{\tau }\right)={\mathfrak{V}}_{\tau }\left(s\right)$), and we know from [18] that

$${\mathfrak{R}}_{{\mathbb{Q}}_s^{\tau }}\left(\xi \right)=m_1^{{\mathbb{Q}}_s^{\tau }}+\mathrm{Var}\left({\mathbb{Q}}_s^{\tau }\right)\xi +\xi \epsilon \left(\xi \right)=s+{\mathfrak{V}}_{\tau }\left(s\right)\xi +\xi \epsilon \left(\xi \right),\epsilon \left(\xi \right)\stackrel{\xi \to 0}{\to }0.$$

(19)

Based on (13) and (19), we get

$${\mathfrak{R}}_{D_{1/t}\left({\mathbb{Q}}_s^{\tau }\right)}\left(\xi \right)={\displaystyle \frac{1}{t}}{\mathfrak{R}}_{{\mathbb{Q}}_s^{\tau }}(\xi /t)={\displaystyle \frac{s}{t}}+{\displaystyle \frac{{\mathfrak{V}}_{\tau }\left(s\right)\xi }{t^2}}+{\displaystyle \frac{\xi }{t^2}}\epsilon \left(\xi \right),$$

(20)

and

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

(21)

Using (12), (15), (20) and (21), from the uniqueness of Taylor expansions, relation (18) becomes

$${\displaystyle \frac{s}{t}}+{\displaystyle \frac{{\mathfrak{V}}_{\tau }\left(s\right)\xi }{t^2}}+{\displaystyle \frac{\xi }{t}}\epsilon \left(\xi \right)=r+{\displaystyle \frac{1}{t}}{\mathfrak{V}}_{\tau }\left(r\right)\xi +\xi {\epsilon }_1\left(\xi \right).$$

(22)

From (22), we get $r=s/t$; then,

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

(23)

We know that the VF is analytic. In addition, for a non-degenerate measure $\tau$, the measures ${\mathbb{Q}}_s^{\tau }$ in ${\mathfrak{F}}_{+}\left(\tau \right)$ are also non-degenerate (for s in the mean domain), and thus their variances ${\mathfrak{V}}_{\tau }\left(s\right)$ are non-zero. Then, relation (23) implies that ${\mathfrak{V}}_{\tau }\left(r\right)=\lambda r$ for $\lambda \ne 0$.

• If $m_1^{\tau }=0$, then there is no VF of the type ${\mathfrak{V}}_{\tau }\left(r\right)=\lambda r$, $\lambda >0$; see [19] (p. 6).
• If $m_1^{\tau }\ne 0$, then $\tau$ represents the image of the MP law (16) via $\zeta \mapsto m_1^{\tau }\zeta$. In this case $\lambda =a^2{m}_1^{\tau }$.

□

Remark 3.

Suppose that $m_1^{\tau }>0$, and τ is the image by $\zeta \mapsto m_1^{\tau }\zeta$ of the MP law (16). In this case, we have $(m_1^{\tau },{\mathit{M}}_{+}^{\tau })=(m_1^{\tau },+\infty ).$ For $t>1$, if $r\in (m_1^{\tau },+\infty )$, then $s=rt\in (t{m}_1^{\tau },+\infty )\subset (m_1^{\tau },+\infty )$. So, relation (23) is well-defined. If $m_1^{\tau }<0$, the one sided mean domain is $(-\infty ,m_1^{\tau })$, and the same conclusion is drawn.

Unfortunately, the inverse implication of (i) is invalid. Assume that $m_1^{\tau }>0$ and that $\tau$ is the image of the MP law (16) by $\zeta \mapsto m_1^{\tau }\zeta$. We show that

$$D_{1/t}\left({\mathbb{Q}}_{rt}^{\tau }\right)\ne {\mathbb{Q}}_r^{D_{1/t}\left({\tau }^{⊞t}\right)}.$$

(24)

We have $m_1^{D_{1/t}\left({\mathbb{Q}}_{rt}^{\tau }\right)}=r=m_1^{{\mathbb{Q}}_r^{D_{1/t}\left({\tau }^{⊞t}\right)}}$. Then, $\varsigma >0$ exists, allowing ${\mathbb{V}}_{D_{1/t}\left({\mathbb{Q}}_{rt}^{\tau }\right)}(·)$ and ${\mathbb{V}}_{{\mathbb{Q}}_r^{D_{1/t}\left({\tau }^{⊞t}\right)}}(·)$ to be defined on $(r,r+\varsigma )$. For $m\in (m_1^{\tau },+\infty )$, we know from [20] (Equation (53)) (for ${\mathbb{V}}_{{\mathbb{Q}}_m^{\tau }}\left(y\right)$ when $\tau$ is MP) that

$${\mathbb{V}}_{{\mathbb{Q}}_m^{\tau }}\left(y\right)=y^2\left({\displaystyle \frac{a^2{m}_1^{\tau }}{y-m}}+{\displaystyle \frac{m_1^{\tau }}{m}}-1\right),y\ne m.$$

(25)

Based on (11) and (25), we obtain

$${\mathbb{V}}_{D_{1/t}\left({\mathbb{Q}}_{rt}^{\tau }\right)}\left(y\right)={\displaystyle \frac{1}{t^2}}{\mathbb{V}}_{{\mathbb{Q}}_{rt}^{\tau }}\left(ty\right)=y^2\left({\displaystyle \frac{a^2{m}_1^{\tau }}{t(y-r)}}+{\displaystyle \frac{m_1^{\tau }}{tr}}-1\right).$$

(26)

Now, we calculate ${\mathbb{V}}_{{\mathbb{Q}}_r^{D_{1/t}\left({\tau }^{⊞t}\right)}}(·)$. We know that

$${\mathbb{V}}_{\tau }\left(u\right)={\displaystyle \frac{a^2{m}_1^{\tau }{u}^2}{u-m_1^{\tau }}}.$$

(27)

From (11), (14) and (27), we get

$${\mathbb{V}}_{D_{1/t}\left({\tau }^{⊞t}\right)}\left(u\right)={\displaystyle \frac{1}{t}}{\mathbb{V}}_{\tau }\left(u\right)={\displaystyle \frac{a^2{m}_1^{\tau }{u}^2}{t(u-m_1^{\tau })}},u\in (m_1^{\tau },+\infty ).$$

(28)

Using [9] (Equation (2.9)), we obtain

$$y={\displaystyle \frac{u^2\left(\frac{a^2{m}_1^{\tau }{r}^2}{t(r-m_1^{\tau })}\right)-r^2\left(\frac{a^2{m}_1^{\tau }{u}^2}{t(u-m_1^{\tau })}\right)}{u\left(\frac{a^2{m}_1^{\tau }{r}^2}{t(r-m_1^{\tau })}\right)-r\left(\frac{a^2{m}_1^{\tau }{u}^2}{t(u-m_1^{\tau })}\right)}}={\displaystyle \frac{ur}{m_1^{\tau }}}.$$

Thus,

$$u={\displaystyle \frac{y{m}_1^{\tau }}{r}}.$$

(29)

Based on [9] (Equation (2.10)), (28), and (29), one has

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

(30)

From (26) and (30), one sees that ${\mathbb{V}}_{{\mathbb{Q}}_r^{D_{1/t}\left({\tau }^{⊞t}\right)}}\left(y\right)\ne {\mathbb{V}}_{D_{1/t}\left({\mathbb{Q}}_{rt}^{\tau }\right)}\left(y\right)$, ∀$y\in (r,r+\varsigma )$. This concludes the proof of (24) using (9).

(ii) Assume that $D_{1/t}\left({\left({\mathfrak{F}}_{+}\left(\tau \right)\right)}^{⊞t}\right)={\mathfrak{F}}_{+}\left(D_{1/t}\left(\tau \right)\right)$. Then, for $s\in (m_1^{\tau },{\mathbf{M}}_{+}^{\tau })$, there exists $r\in (m_1^{D_{1/t}\left(\tau \right)},{\mathbf{M}}_{+}^{D_{1/t}\left(\tau \right)})=({\displaystyle \frac{m_1^{\tau }}{t}},{\displaystyle \frac{{\mathbf{M}}_{+}^{\tau }}{t}})$ so that

$$D_{1/t}{\left({\mathbb{Q}}_s^{\tau }\right)}^{⊞t}={\mathbb{Q}}_r^{D_{1/t}\left(\tau \right)}.$$

(31)

That is,

$${\mathfrak{R}}_{D_{1/t}{\left({\mathbb{Q}}_s^{\tau }\right)}^{⊞t}}\left(\xi \right)={\mathfrak{R}}_{{\mathbb{Q}}_r^{D_{1/t}\left(\tau \right)}}\left(\xi \right),\xi \mathrm{close} \mathrm{to}0.$$

(32)

This implies that

$$m_1^{D_{1/t}{\left({\mathbb{Q}}_s^{\tau }\right)}^{⊞t}}+\mathrm{Var}\left(D_{1/t}{\left({\mathbb{Q}}_s^{\tau }\right)}^{⊞t}\right)\xi +\xi \epsilon \left(\xi \right)=m_1^{{\mathbb{Q}}_r^{D_{1/t}\left(\tau \right)}}+\mathrm{Var}\left({\mathbb{Q}}_r^{D_{1/t}\left(\tau \right)}\right)\xi +\xi {\epsilon }_2\left(\xi \right).$$

Equivalently,

$$s+{\displaystyle \frac{{\mathfrak{V}}_{\tau }\left(s\right)}{t}}\xi +\xi \epsilon \left(\xi \right)=r+{\displaystyle \frac{{\mathfrak{V}}_{\tau }\left(rt\right)}{t^2}}+\xi {\epsilon }_2\left(\xi \right),\epsilon \left(\xi \right)\stackrel{\xi \to 0}{\to }0\mathrm{and}{\epsilon }_2\left(\xi \right)\stackrel{\xi \to 0}{\to }0.$$

(33)

It is clear from (33) that $s=r$; then,

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

(34)

As in (i), the same conclusion is provided: That is, $m_1^{\tau }\ne 0$, and $\tau$ represents the image of the MP law (16) via $\zeta \mapsto m_1^{\tau }\zeta$.

Remark 4.

Suppose that $m_1^{\tau }>0$. For $t>1$, if $s\in (m_1^{\sigma },+\infty )$, then $r=s\in (m_1^{\sigma },+\infty )\subset ({\displaystyle \frac{m_1^{\tau }}{t}},+\infty )$. Thus, relation (34) is well-defined. The same conclusion is drawn if $m_1^{\sigma }<0$.

The inverse implication of (ii) is invalid. That is,

$$D_{1/t}{\left({\mathbb{Q}}_r^{\tau }\right)}^{⊞t}\ne {\mathbb{Q}}_r^{D_{1/t}\left(\tau \right)}.$$

(35)

We have that $m_1^{D_{1/t}{\left({\mathbb{Q}}_r^{\tau }\right)}^{⊞t}}=r=m_1^{{\mathbb{Q}}_r^{D_{1/t}\left(\tau \right)}}$. Then, $\iota >0$ exists so that ${\mathbb{V}}_{D_{1/t}{\left({\mathbb{Q}}_r^{\tau }\right)}^{⊞t}}(·)$ and ${\mathbb{V}}_{{\mathbb{Q}}_r^{D_{1/t}\left(\tau \right)}}(·)$ are defined on $(r,r+\iota )$. Using (11), (14), and (25), we get

$${\mathbb{V}}_{D_{1/t}{\left({\mathbb{Q}}_r^{\tau }\right)}^{⊞t}}\left(y\right)={\displaystyle \frac{1}{t}}{\mathbb{V}}_{{\mathbb{Q}}_r^{\tau }}\left(y\right)={\displaystyle \frac{y^2}{t}}\left({\displaystyle \frac{a^2{m}_1^{\tau }}{y-r}}+{\displaystyle \frac{m_1^{\tau }}{r}}-1\right),y\in (r,r+\iota ).$$

(36)

We calculate ${\mathbb{V}}_{{\mathbb{Q}}_r^{D_{1/t}\left(\tau \right)}}(·)$. From (11) and (27), we get

$${\mathbb{V}}_{D_{1/t}\left(\tau \right)}\left(m\right)={\displaystyle \frac{1}{t^2}}{\mathbb{V}}_{\tau }\left(tm\right)={\displaystyle \frac{a^2{m}_1^{\tau }{m}^2}{tm-m_1^{\tau }}},m\in ({\displaystyle \frac{m_1^{\tau }}{t}},+\infty ).$$

(37)

Using [9] (Equation (2.9)), we get

$$y={\displaystyle \frac{m^2\left(\frac{a^2{r}^2}{(rt-m_1^{\tau })}\right)-r^2\left(\frac{a^2{m}^2}{(mt-m_1^{\tau })}\right)}{m\left(\frac{a^2{r}^2}{(rt-m_1^{\tau })}\right)-r\left(\frac{a^2{m}^2}{(tm-m_1^{\tau })}\right)}}={\displaystyle \frac{tmr}{m_1^{\tau }}}.$$

Then,

$$m={\displaystyle \frac{y{m}_1^{\tau }}{tr}}.$$

(38)

Based on [9] (Equation (2.10)), (37), and (38), we obtain

$${\mathbb{V}}_{{\mathbb{Q}}_r^{D_{1/t}\left(\tau \right)}}\left(y\right)=y\left({\displaystyle \frac{{\mathbb{V}}_{D_{1/t}\left(\tau \right)}\left(m\right)}{m}}+m-y\right)=y^2\left({\displaystyle \frac{a^2}{t(y-r)}}+{\displaystyle \frac{m_1^{\tau }}{tr}}-1\right),y\ne r.$$

(39)

One see from (36) and (39) that ${\mathbb{V}}_{D_{1/t}{\left({\mathbb{Q}}_r^{\tau }\right)}^{⊞t}}\left(y\right)\ne {\mathbb{V}}_{{\mathbb{Q}}_r^{D_{1/t}\left(\tau \right)}}\left(y\right)$, ∀$y\in (r,r+\iota )$. This concludes the proof of (35) using (9).

(iii) Assume that $D_{1/t}\left({\mathfrak{F}}_{+}\left({\tau }^{⊞t}\right)\right)={\mathfrak{F}}_{+}\left(D_{1/t}\left(\tau \right)\right)$. Then, for $s\in (m_1^{{\tau }^{⊞t}},{\mathbf{M}}_{+}^{{\tau }^{⊞t}})=(t{m}_1^{\tau },t{\mathbf{M}}_{+}^{\tau })$, there exists $r\in (m_1^{D_{1/t}\left(\tau \right)},{\mathbf{M}}_{+}^{D_{1/t}\left(\tau \right)})=({\displaystyle \frac{m_1^{\tau }}{t}},{\displaystyle \frac{{\mathbf{M}}_{+}^{\tau }}{t}})$, so that

$$D_{1/t}\left({\mathbb{Q}}_s^{{\tau }^{⊞t}}\right)={\mathbb{Q}}_r^{D_{1/t}\left(\tau \right)}.$$

(40)

That is,

$${\mathfrak{R}}_{D_{1/t}\left({\mathbb{Q}}_s^{{\tau }^{⊞t}}\right)}\left(\xi \right)={\mathfrak{R}}_{{\mathbb{Q}}_r^{D_{1/t}\left(\tau \right)}}\left(\xi \right),\xi \mathrm{close} \mathrm{to}0.$$

(41)

Relation (41) may be written as

$${\displaystyle \frac{s}{t}}+{\displaystyle \frac{1}{t}}{\mathfrak{V}}_{\tau }(s/t)\xi +\xi \epsilon \left(\xi \right)=r+{\displaystyle \frac{1}{t^2}}{\mathfrak{V}}_{\tau }\left(tr\right)\xi +\xi {\epsilon }_3\left(\xi \right),\epsilon \left(\xi \right)\stackrel{\xi \to 0}{\to }0\mathrm{and}{\epsilon }_3\left(\xi \right)\stackrel{\xi \to 0}{\to }0.$$

(42)

It is clear from (42) that $r=s/t$; then,

$${\mathfrak{V}}_{\tau }(s/t)={\mathfrak{V}}_{\tau }\left(s\right)/t,\forall s\in (t{m}_1^{\tau },t{\mathbf{M}}_{+}^{\tau }),\mathrm{and}\forall t>1.$$

(43)

Then, $m_1^{\tau }\ne 0$ and $\tau$ represents the image of the MP law (16) via $\zeta \mapsto m_1^{\tau }\zeta$.

Remark 5.

Suppose that $m_1^{\tau }>0$. For $t>1$, if $s\in (t{m}_1^{\tau },+\infty )$, then $r=s/t\in (m_1^{\tau },+\infty )\subset ({\displaystyle \frac{m_1^{\tau }}{t}},+\infty )$. Thus, relation (43) is well-defined. The same conclusion is drawn if $m_1^{\sigma }<0$.

The inverse implication of (iii) is invalid. That is,

$$D_{1/t}\left({\mathbb{Q}}_{rt}^{{\tau }^{⊞t}}\right)\ne {\mathbb{Q}}_r^{D_{1/t}\left(\tau \right)}.$$

(44)

We have that $m_1^{D_{1/t}\left({\mathbb{Q}}_{rt}^{{\tau }^{⊞t}}\right)}=r=m_1^{{\mathbb{Q}}_r^{D_{1/t}\left(\tau \right)}}$. Then, $\varsigma >0$ exists ensuring that ${\mathbb{V}}_{D_{1/t}\left({\mathbb{Q}}_{rt}^{{\tau }^{⊞t}}\right)}(·)$ and ${\mathbb{V}}_{{\mathbb{Q}}_r^{D_{1/t}\left(\tau \right)}}(·)$ are defined on $(r,r+\varsigma )$. We know from [20] (Equation (59)) that

$${\mathbb{V}}_{{\mathbb{Q}}_r^{{\tau }^{⊞t}}}\left(y\right)=y^2\left({\displaystyle \frac{a^2{m}_1^{\tau }}{y-r}}+{\displaystyle \frac{t{m}_1^{\tau }}{r}}-1\right),y\ne r.$$

(45)

Combining (11) and (45), we obtain

$${\mathbb{V}}_{D_{1/t}\left({\mathbb{Q}}_{rt}^{{\tau }^{⊞t}}\right)}\left(y\right)=y^2\left({\displaystyle \frac{a^2{m}_1^{\tau }}{t(y-r)}}+{\displaystyle \frac{m_1^{\tau }}{r}}-1\right)y\ne r.$$

(46)

One sees from (39) and (46) that ${\mathbb{V}}_{D_{1/t}\left({\mathbb{Q}}_{rt}^{{\tau }^{⊞t}}\right)}(·)\left(y\right)\ne {\mathbb{V}}_{{\mathbb{Q}}_r^{D_{1/t}\left(\tau \right)}}\left(y\right)$, ∀$y\in (r,r+\varsigma )$. This ends the proof of (44) by the use of (9).

(iv) Assume that ${\mathfrak{F}}_{+}\left({\tau }^{⊞t}\right)={\mathfrak{F}}_{+}\left(\tau \right)$. Then, for $s\in (m_1^{{\tau }^{⊞t}},{\mathbf{M}}_{+}^{{\tau }^{⊞t}})=(t{m}_1^{\tau },t{\mathbf{M}}_{+}^{\tau })$, there exists $r\in (m_1^{\tau },{\mathbf{M}}_{+}^{\tau })$, so that

$${\mathbb{Q}}_s^{{\tau }^{⊞t}}={\mathbb{Q}}_r^{\tau }.$$

(47)

That is,

$${\mathfrak{R}}_{{\mathbb{Q}}_s^{{\tau }^{⊞t}}}\left(\xi \right)={\mathfrak{R}}_{{\mathbb{Q}}_r^{\tau }}\left(\xi \right),\xi \mathrm{close} \mathrm{to}0.$$

(48)

Relation (48) may be written as

$$s+t{\mathfrak{V}}_{\tau }(s/t)\xi +\xi \epsilon \left(\xi \right)=r+{\mathfrak{V}}_{\tau }\left(r\right)\xi +\xi \epsilon \left(\xi \right),\epsilon \left(\xi \right)\stackrel{\xi \to 0}{\to }0.$$

(49)

It is clear from (49) that $r=s$; then,

$${\mathfrak{V}}_{\tau }(s/t)={\mathfrak{V}}_{\tau }\left(s\right)/t,\forall s\in (t{m}_1^{\tau },t{\mathbf{M}}_{+}^{\tau }),\mathrm{and}\forall t>1.$$

(50)

Then, $m_1^{\tau }\ne 0$, and $\tau$ represents the image of the MP law (16) via $\zeta \mapsto m_1^{\tau }\zeta$.

Remark 6.

Suppose that $m_1^{\tau }>0$. For $t>1$, if $s\in (t{m}_1^{\tau },+\infty )$, then $r=s\in (t{m}_1^{\tau },+\infty )\subset (m_1^{\tau },+\infty )$. Thus, relation (50) is well-defined. The same conclusion is drawn if $m_1^{\sigma }<0$.

The inverse implication of (iv) is also invalid. That is,

$${\mathbb{Q}}_r^{{\tau }^{⊞t}}\ne {\mathbb{Q}}_r^{\tau }.$$

(51)

We have that $m_1^{{\mathbb{Q}}_r^{{\tau }^{⊞t}}}=r=m_1^{{\mathbb{Q}}_r^{\tau }}$. Then, $\iota >0$ exists so that ${\mathbb{V}}_{{\mathbb{Q}}_r^{{\tau }^{⊞t}}}(·)$ and ${\mathbb{V}}_{{\mathbb{Q}}_r^{\tau }}(·)$ are well-defined on $(r,r+\iota )$. From (25) and (45), one sees that ${\mathbb{V}}_{{\mathbb{Q}}_r^{{\tau }^{⊞t}}}\left(y\right)\ne {\mathbb{V}}_{{\mathbb{Q}}_r^{\tau }}\left(y\right)$, ∀$y\in (r,r+\iota )$. This concludes the proof of (51) using (9).

3. Notes on the FG Law

For $a\ne 0$, The FG law is provided by

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

(52)

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

The FG law serves the same function in non-commutative probability that the Gamma law did in classical probability. It has received much attention from researchers in recent decades. In [21], the authors established some properties of the density related to the FG law, including analyticity, unimodality, and the asymptotic behavior. In [7], being a member of the free Meixner family, the FG law is described by the associated VF. In this part, we discuss additional features of the FG law in relation to CSK families, based on the ideas of free additive convolution and dilation of measures. More specifically, we have the following:

Theorem 2.

Let $\tau \in {\mathcal{P}}_{ba}$, with the finite first moment $m_1^{\tau }$. For $t>1$, consider the sets of measures defined by (3) and (5). If

(i) 

${\left({\mathfrak{F}}_{+}\left(\tau \right)\right)}^{⊞t}={\mathfrak{F}}_{+}\left(D_{1/t}\left({\tau }^{⊞t}\right)\right)$ or

(ii) 

$D_{1/t}\left({\left({\mathfrak{F}}_{+}\left(\tau \right)\right)}^{⊞t}\right)={\mathfrak{F}}_{+}\left({\tau }^{⊞t}\right)$ or

(iii) 

${\mathfrak{F}}_{+}\left(D_{1/t}\left(\tau \right)\right)={\mathfrak{F}}_{+}\left(\tau \right)$

then τ is the FG law (52) up to scaling.

Proof. 

(i) Assume that ${\left({\mathfrak{F}}_{+}\left(\tau \right)\right)}^{⊞t}={\mathfrak{F}}_{+}\left(D_{1/t}\left({\tau }^{⊞t}\right)\right)$. Then, for $l\in (m_1^{\tau },{\mathbf{M}}_{+}^{\tau })$, there exists $r\in (m_1^{D_{1/t}\left({\tau }^{⊞t}\right)},{\mathbf{M}}_{+}^{D_{1/t}\left({\tau }^{⊞t}\right)})=(m_1^{\tau },{\mathbf{M}}_{+}^{\tau })$ so that

$${\left({\mathbb{Q}}_l^{\tau }\right)}^{⊞t}={\mathbb{Q}}_r^{D_{1/t}\left({\tau }^{⊞t}\right)}.$$

(53)

That is,

$$t{\mathfrak{R}}_{{\mathbb{Q}}_l^{\tau }}\left(\xi \right)={\mathfrak{R}}_{{\mathbb{Q}}_r^{D_{1/t}\left({\tau }^{⊞t}\right)}}\left(\xi \right),\xi \mathrm{close} \mathrm{to}0.$$

(54)

As $m_1^{\sigma }$ is finite, we know from [18] that

$${\mathfrak{R}}_{{\mathbb{Q}}_l^{\tau }}\left(\xi \right)=l+{\mathfrak{V}}_{\tau }\left(l\right)\xi +\xi \epsilon \left(\xi \right),\epsilon \left(\xi \right)\stackrel{\xi \to 0}{\to }0.$$

(55)

Additionally, we have

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

(56)

Based on (12), (15), (55), and (56), relation (54) becomes

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

(57)

It is clear from (57) that $r=tl$; then,

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

(58)

${\mathfrak{V}}_{\tau }(·)\ne 0$ as $\tau$ is non-degenerate. Thus, ${\mathfrak{V}}_{\tau }\left(l\right)=\lambda l^2$ for $\lambda >0$.

• If $m_1^{\tau }=0$, then there is no VF of the type $V\left(l\right)=\lambda l^2$, where $\lambda >0$. See [20].
• If $m_1^{\tau }\ne 0$, then $\tau$ is the image of the FG law (52) via $\zeta \mapsto m_1^{\tau }\zeta$. In this case, $\lambda =a^2$.

□

Remark 7.

Suppose that $m_1^{\tau }>0$. In that case, we have $(m_1^{\tau },{\mathit{M}}_{+}^{\tau })=(m_1^{\tau },+\infty ).$ For $t>1$, if $l\in (m_1^{\tau },+\infty )$, then $r=tl\in (t{m}_1^{\tau },+\infty )\subset (m_1^{\tau },+\infty )$. So, relation (58) is well-defined. If $m_1^{\tau }<0$, we deal with left-sided CSK families. In this case, the one-sided mean domain is $(-\infty ,m_1^{\tau })$, and the same conclusion is drawn.

The inverse implication of (i) is invalid. Assume that $m_1^{\tau }>0$, and $\tau$ is the image by $\zeta \mapsto m_1^{\tau }\zeta$ of ${\mathbf{fg}}_{\left(a\right)}$ given by (52). We show that

$${\left({\mathbb{Q}}_l^{\tau }\right)}^{⊞t}\ne {\mathbb{Q}}_{tl}^{D_{1/t}\left({\tau }^{⊞t}\right)}.$$

(59)

We have that $m_1^{{\left({\mathbb{Q}}_l^{\tau }\right)}^{⊞t}}=tl=m_1^{{\mathbb{Q}}_{tl}^{D_{1/t}\left({\tau }^{⊞t}\right)}}$. Then, there exists $\varsigma >0$ ensuring that ${\mathbb{V}}_{{\left({\mathbb{Q}}_l^{\tau }\right)}^{⊞t}}(·)$ and ${\mathbb{V}}_{{\mathbb{Q}}_{tl}^{D_{1/t}\left({\tau }^{⊞t}\right)}}(·)$ are defined on $(tl,tl+\varsigma )$. From [20] (Equation (41)), (for ${\mathbb{V}}_{{\mathbb{Q}}_l^{\tau }}\left(y\right)$ when $\tau$ is FG), we have

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

(60)

Based on (14) and (60), we obtain

$${\mathbb{V}}_{{\left({\mathbb{Q}}_l^{\tau }\right)}^{⊞t}}\left(y\right)={\displaystyle \frac{y^3(a^2{l}^2-(y/t-l)(l-m_1^{\tau }))}{(y-tl)(y(l-m_1^{\tau })+tl{m}_1^{\tau })}},y\ne tl.$$

(61)

Now, we calculate ${\mathbb{V}}_{{\mathbb{Q}}_r^{D_{1/t}\left({\tau }^{⊞t}\right)}}(·)$. For $u\in (m_1^{\tau },+\infty )$, we have

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

(62)

Using [9] (Equation (2.9)), we obtain

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

Thus,

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

(63)

Using [9] (Equation (2.10)), (62), and (63), we obtain

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

(64)

From (61) and (64), it is clear that ${\mathbb{V}}_{{\mathbb{Q}}_{tl}^{D_{1/t}\left({\tau }^{⊞t}\right)}}\left(x\right)\ne {\mathbb{V}}_{{\left({\mathbb{Q}}_l^{\tau }\right)}^{⊞t}}\left(x\right)$, ∀$x\in (tl,tl+\varsigma )$. This concludes the proof of (59) based on (9).

(ii) Assume that $D_{1/t}\left({\left({\mathfrak{F}}_{+}\left(\tau \right)\right)}^{⊞t}\right)={\mathfrak{F}}_{+}\left({\tau }^{⊞t}\right)$. Then, for $l\in (m_1^{{\tau }^{⊞t}},{\mathbf{M}}_{+}^{{\tau }^{⊞t}})=(t{m}_1^{\tau },t{\mathbf{M}}_{+}^{\tau })$, there exists $s\in (m_1^{\tau },{\mathbf{M}}_{+}^{\tau })$ so that

$${\mathbb{Q}}_l^{{\tau }^{⊞t}}=D_{1/t}\left({\left({\mathbb{Q}}_s^{\tau }\right)}^{⊞t}\right).$$

(65)

That is,

$${\mathfrak{R}}_{{\mathbb{Q}}_l^{{\tau }^{⊞t}}}\left(\xi \right)={\mathfrak{R}}_{D_{1/t}\left({\left({\mathbb{Q}}_s^{\tau }\right)}^{⊞t}\right)}\left(\xi \right),\xi \mathrm{close} \mathrm{to}0.$$

(66)

This implies that

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

(67)

It is clear from (67) that $s=l$; then,

$${\mathfrak{V}}_{\tau }(l/t)={\mathfrak{V}}_{\tau }\left(l\right)/t^2,\forall l\in (t{m}_1^{\tau },t{\mathbf{M}}_{+}^{\tau }),\mathrm{and}\forall t>1.$$

(68)

As in (i), the same conclusion is provided. That is, $m_1^{\tau }\ne 0$, and $\tau$ is the image of the FG law (52) via $\zeta \mapsto m_1^{\tau }\zeta$.

Remark 8.

Suppose that $m_1^{\tau }>0$. In that case, we have $(m_1^{{\tau }^{⊞t}},{\mathit{M}}_{+}^{{\tau }^{⊞t}})=(t{m}_1^{\tau },+\infty )$. For $t>1$, if $l\in (t{m}_1^{\tau },+\infty )$, then $s=l\in (t{m}_1^{\tau },+\infty )\subset (m_1^{\tau },+\infty )$. Thus, relation (68) is well-defined. The same conclusion is drawn if $m_1^{\tau }<0$.

The opposite implication in (ii) is also invalid. That is,

$$D_{1/t}\left({\left({\mathbb{Q}}_l^{\tau }\right)}^{⊞t}\right)\ne {\mathbb{Q}}_l^{\left({\tau }^{⊞t}\right)}.$$

(69)

We have that $m_1^{D_{1/t}\left({\left({\mathbb{Q}}_l^{\tau }\right)}^{⊞t}\right)}=l=m_1^{{\mathbb{Q}}_l^{\left({\tau }^{⊞t}\right)}}$. Then, $\iota >0$ exists so that ${\mathbb{V}}_{D_{1/t}\left({\left({\mathbb{Q}}_l^{\tau }\right)}^{⊞t}\right)}(·)$ and ${\mathbb{V}}_{{\mathbb{Q}}_l^{\left({\tau }^{⊞t}\right)}}(·)$ are well-defined on $(l,l+\iota )$. From (11), (14), and (60), we have

$${\mathbb{V}}_{D_{1/t}\left({\left({\mathbb{Q}}_l^{\tau }\right)}^{⊞t}\right)}\left(y\right)={\displaystyle \frac{1}{t}}{\mathbb{V}}_{{\mathbb{Q}}_l^{\tau }}\left(y\right)={\displaystyle \frac{1}{t}}{\displaystyle \frac{y^3(a^2{l}^2-(y-l)(l-m_1^{\tau }))}{(y-l)(y(l-m_1^{\tau })+l{m}_1^{\tau })}},y\ne l.$$

(70)

We calculate ${\mathbb{V}}_{{\mathbb{Q}}_l^{\left({\tau }^{⊞t}\right)}}(·)$. We have

$${\mathbb{V}}_{{\tau }^{⊞t}}\left(m\right)=t{\mathbb{V}}_{\tau }(m/t)={\displaystyle \frac{1}{t}}{\displaystyle \frac{a^2{m}^3}{m-t{m}_1^{\tau }}},m\in (t{m}_1^{\tau },+\infty ).$$

(71)

Using [9] (Equation (2.9)), we get

$$w={\displaystyle \frac{m^2\left(\frac{a^2{l}^3}{t(l-t{m}_1^{\tau })}\right)-l^2\left(\frac{a^2{m}^3}{t(m-t{m}_1^{\tau })}\right)}{m\left(\frac{a^2{l}^3}{t(l-t{m}_1^{\tau })}\right)-l\left(\frac{a^2{m}^3}{t(m-t{m}_1^{\tau })}\right)}}={\displaystyle \frac{mlt{m}_1^{\tau }}{t{m}_1^{\tau }(m+l)-ml}}.$$

Then,

$$m={\displaystyle \frac{wlt{m}_1^{\tau }}{lw+t{m}_1^{\tau }l-t{m}_1^{\tau }w}}.$$

(72)

Based on [9] (Equation (2.10)), (71), and (72), we obtain

$${\mathbb{V}}_{{\mathbb{Q}}_l^{{\tau }^{⊞t}}}\left(w\right)=w\left({\displaystyle \frac{{\mathbb{V}}_{{\tau }^{⊞t}}\left(m\right)}{m}}+m-w\right)={\displaystyle \frac{w^3(a^2{l}^2-t(w-l)(l-t{m}_1^{\tau }))}{t(w-l)(lw-wt{m}_1^{\tau }+lt{m}_1^{\tau })}},w\ne l.$$

(73)

One sees from (70) and (73) that ${\mathbb{V}}_{{\mathbb{Q}}_l^{{\tau }^{⊞t}}}\left(x\right)\ne {\mathbb{V}}_{D_{1/t}\left({\left({\mathbb{Q}}_l^{\tau }\right)}^{⊞t}\right)}\left(x\right)$, ∀$x\in (l,l+\iota )$. This ends the proof of (69).

(iii) Assume that ${\mathfrak{F}}_{+}\left(D_{1/t}\left(\tau \right)\right)={\mathfrak{F}}_{+}\left(\tau \right)$. Then, for $l\in (m_1^{\tau },{\mathbf{M}}_{+}^{\tau })$, there exists $p\in (m_1^{D_{1/t}\left(\tau \right)},{\mathbf{M}}_{+}^{D_{1/t}\left(\tau \right)})=({\displaystyle \frac{m_1^{\tau }}{t}},{\displaystyle \frac{{\mathbf{M}}_{+}^{\tau }}{t}})$ so that

$${\mathbb{Q}}_l^{\tau }={\mathbb{Q}}_p^{D_{1/t}\left(\tau \right)}.$$

(74)

That is,

$${\mathfrak{R}}_{{\mathbb{Q}}_l^{\tau }}\left(\xi \right)={\mathfrak{R}}_{{\mathbb{Q}}_p^{D_{1/t}\left(\tau \right)}}\left(\xi \right),\xi \mathrm{close} \mathrm{to} 0.$$

(75)

This gives

$$l+{\mathfrak{V}}_{\tau }\left(l\right)\xi +\xi \epsilon \left(\xi \right)=p+{\displaystyle \frac{1}{t^2}}{\mathfrak{V}}_{\tau }\left(tp\right)\xi +\xi {\epsilon }_3\left(\xi \right),\epsilon \left(\xi \right)\stackrel{\xi \to 0}{\to }0\mathrm{and}{\epsilon }_3\left(\xi \right)\stackrel{\xi \to 0}{\to }0.$$

(76)

It is clear from (76) that $p=l$; then,

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

(77)

Then, $m_1^{\tau }\ne 0$, and $\tau$ is the image of the FG law (52) via $\zeta \mapsto m_1^{\tau }\zeta$.

Remark 9.

Suppose that $m_1^{\tau }>0$. In that case, we have $(m_1^{D_{1/t}\left(\tau \right)},{\mathit{M}}_{+}^{D_{1/t}\left(\tau \right)})=({\displaystyle \frac{m_1^{\tau }}{t}},{\displaystyle \frac{{\mathit{M}}_{+}^{\tau }}{t}})=({\displaystyle \frac{m_1^{\tau }}{t}},+\infty )$. For $t>1$, if $l\in (m_1^{\tau },+\infty )$, then $p=l\in (m_1^{\tau },+\infty )\subset ({\displaystyle \frac{m_1^{\tau }}{t}},+\infty )$. Thus, relation (77) is well-defined. The same conclusion is drawn if $m_1^{\tau }<0$.

The opposite implication of (iii) is also invalid. That is,

$${\mathbb{Q}}_l^{D_{1/t}\left(\tau \right)}\ne {\mathbb{Q}}_l^{\tau }.$$

(78)

For $v\in ({\displaystyle \frac{m_1^{\tau }}{t}},+\infty )$, we have

$${\mathbb{V}}_{D_{1/t}\left(\tau \right)}\left(v\right)={\displaystyle \frac{a^{2}t{v}^3}{tv-m_1^{\tau }}}.$$

(79)

Using [9] Equations (2.9) and (79), we get

$$y={\displaystyle \frac{v^2\left(\frac{a^{2}t{l}^3}{tl-m_1^{\tau }}\right)-l^2\left(\frac{a^{2}t{v}^3}{tv-m_1^{\tau }}\right)}{v\left(\frac{a^{2}t{l}^3}{tl-m_1^{\tau }}\right)-l\left(\frac{a^{2}t{v}^3}{tv-m_1^{\tau }}\right)}}={\displaystyle \frac{vl{m}_1^{\tau }}{m_1^{\tau }(v+l)-tvl}}.$$

(80)

From (80), we have

$$v={\displaystyle \frac{yl{m}_1^{\tau }}{tyl+l{m}_1^{\tau }-y{m}_1^{\tau }}}.$$

(81)

Relations [9] (Equation (2.10)), (79), and (81) give

$${\mathbb{V}}_{{\mathbb{Q}}_l^{D_{1/t}\left(\tau \right)}}\left(y\right)=y\left({\displaystyle \frac{{\mathbb{V}}_{D_{1/t}\left(\tau \right)}\left(v\right)}{v}}+v-y\right)={\displaystyle \frac{y^3(a^{2}t{l}^2-(y-l)(tl-m_1^{\tau }))}{(y-l)(y(tl-m_1^{\tau })+l{m}_1^{\tau })}},y\ne l.$$

(82)

Relation (78) follows by comparing (60) and (82).

4. Conclusions

Studying the relationships between CSK families in terms of probability measures, convolutions, and dilations is important because these operations reflect how distributions behave under combination and scaling in probability theory. Kernel families provide a unified analytic framework, where convolution corresponds to a form of independence (classical, free, Boolean, etc.), and dilation reveals scaling properties and self-similarity. These interactions are central to understanding limit laws, infinite divisibility, and stable distributions. Moreover, such structures appear naturally in free probability and random matrix theory, where they relate to deep conjectures about universality and spectral behavior. By highlighting these connections, this paper can situate its results within a broader context of ongoing research into the algebraic and analytic foundations of modern probability where identifying fixed points or characterizing semigroup structures remains an active area of investigation.

This article investigates some features of the MP and FG laws, concentrating on their links to CSK families, based on free additive convolution and measure dilations. Our exploration highlights the fundamental connections between free probability and analytic function theory, demonstrating the utility of CSK families in advancing the theoretical and computational aspects of free additive convolution. The results presented not only deepen our understanding of spectral distributions but also open avenues for further research, including extensions to non-traditional probability spaces, asymptotic approximations, and applications in high-dimensional statistics. In fact, while this work’s primary focus is theoretical, structural insights into CSK families, free convolution, and dilation may feed numerical and algorithmic techniques in adjacent fields. Understanding how specific measures respond under free convolution and scaling, for example, might help to create fast algorithms for simulating free random variables or estimating their spectral distributions, both of which are important in random matrix theory and high-dimensional statistics. Furthermore, the characterization of fixed points and stability under transformations may be useful in developing iterative techniques or optimization routines in non-commutative environments, where equivalents of traditional probabilistic algorithms are still being developed. While these paths are outside the scope of the current paper, they provide potential opportunities for future study at the intersection of free probability and computing.

While the analogy between the MP (respectively, FG) law in free probability and the Poisson (respectively, Gamma) law in classical probability is conceptually useful, especially in their roles as limit distributions for sums of freely or classically independent random variables, the characterizations developed in this work rely on tools and structures that do not have classical counterparts. Specifically, classical approaches often characterize distributions like the Poisson or Gamma via properties of NEFs, such as VFs, conjugate priors, or operations like classical convolution. In contrast, the characterizations presented here for the MP and FG laws are rooted in the non-commutative framework of free probability, involving CSK families, free additive convolution, and scaling operations. This raises an intriguing question: could the types of structural and stability properties explored here for free families have meaningful analogues within the classical NEF setting? For example, are there classical exponential families whose behavior under convolution and dilation mirrors, even abstractly, the fixed-point or invariance phenomena seen in free probability? Exploring such parallels could shed light on deeper connections between classical and free probabilistic structures and possibly lead to new characterizations of classical NEFs from a transformational or functional–analytic perspective.