A Two-Parameter Extension of the V_a Deformation of Probability Measures

Abstract

This article proposes a single-deformation scheme for probability measures that simultaneously encompasses two classical deformations from free probability: the $V_a$ deformation ($a\in \mathbb{R}$) and the $T_c$ deformation ($c\in \mathbb{R}$). The associated operator, written as ${\mathcal{W}}_{(a,c)}$, is introduced via a functional relation involving the Cauchy–Stieltjes transform and is constructed so as to recover the initial deformations as special cases, namely ${\mathcal{W}}_{(0,c)}=T_c$ and ${\mathcal{W}}_{(a,0)}=V_a$. Working within the concept of Cauchy–Stieltjes kernel families, we analyze the action of ${\mathcal{W}}_{(a,c)}$ on variance functions and establish an explicit expression for the variance function induced by this deformation. This approach leads to a structural invariance property demonstrating that the free Meixner class is preserved under the action of ${\mathcal{W}}_{(a,c)}$. In addition, the operator provides a new perspective on the semicircle distribution, yielding a characterization that reflects the symmetric nature of the deformation and its compatibility with fundamental distributions in free probability.

1. Introduction

In free probability theory, deformations of probability measures provide an effective framework for understanding how convolution structures interact with analytic properties of distributions. Such deformations play an important role in the investigation of stability, infinite divisibility, and limit theorems in noncommutative probability [1,2,3,4,5]. Among the deformations that have been widely studied are the $V_a$ deformation introduced in [6] and the $T_c$ deformation considered in [7,8]. These operators modify probability measures through analytic transformations of their Cauchy–Stieltjes transforms (CSTs) and have proven useful in the analysis of Cauchy–Stieltjes kernel (CSK) families and the related variance functions (VFs). Despite their similarities, the $V_a$ and $T_c$ deformations have mostly been investigated independently. This separation obscures structural relations between the two operators and limits a unified view of measure deformations in free probability. Motivated by this observation, we introduce a two-parameter deformation that combines both mechanisms into a single framework. The proposed operator, denoted as ${\mathcal{W}}_{(a,c)}$, extends the classical $V_a$ and $T_c$ deformations and allows for a unified analysis of their analytic and probabilistic effects.

Let $\mathbf{P}$ (resp. ${\mathbf{P}}_c$) denote the set of probability measures on $\mathbb{R}$ (resp. with compact support). The CST for $\lambda \in \mathbf{P}$ is defined by

$${\mathbf{G}}_{\lambda }\left(w\right)=\int {\displaystyle \frac{\lambda \left(dl\right)}{w-l}},w\in \mathbb{C}\setminus supp\left(\lambda \right).$$

The $V_a$ deformation introduced in [6] is defined for measures with finite variance by

$${\displaystyle \frac{1}{{\mathbf{G}}_{V_a\left(\lambda \right)}\left(w\right)}}={\displaystyle \frac{1}{{\mathbf{G}}_{\lambda }\left(w\right)}}+a{\mathbf{r}}_2\left(\lambda \right),$$

(1)

where $a\in \mathbb{R}$ and ${\mathbf{r}}_2\left(\lambda \right)$ denotes the variance (the second free cumulant) of $\lambda$. The function expressed as ${\mathbf{G}}_{V_a\left(\lambda \right)}(·)$ is the CST of a unique probability measure ($V_a\left(\lambda \right)$). Another deformation acting on the CST was studied in [7,8]. For $\lambda \in \mathbf{P}$ and $c\in \mathbb{R}$, it is defined by

$${\displaystyle \frac{1}{{\mathbf{G}}_{T_c\left(\lambda \right)}\left(w\right)}}={\displaystyle \frac{1}{{\mathbf{G}}_{\lambda }(w-c)}}+c.$$

(2)

Again, the function expressed as ${\mathbf{G}}_{T_c\left(\lambda \right)}(·)$ is the CST of a uniquely determined probability measure ($T_c\left(\lambda \right)$). This deformation introduces a controlled shift in the analytic argument of the CST and naturally appears in investigations of perturbations of free convolution structures.

The similarity of the analytic forms of these two operators suggests the possibility of a unified construction. To this end, we introduce the following two-parameter deformation. For $\lambda \in \mathbf{P}$ with a finite second moment and parameters a, $c\in \mathbb{R}$, we introduce

$${\displaystyle \frac{1}{{\mathbf{G}}_{{\mathcal{W}}_{(a,c)}\left(\lambda \right)}\left(w\right)}}={\displaystyle \frac{1}{{\mathbf{G}}_{\lambda }(w-c)}}+a{\mathbf{r}}_2\left(\lambda \right)+c.$$

(3)

The function expressed as ${\mathbf{G}}_{{\mathcal{W}}_{(a,c)}\left(\lambda \right)}(·)$ is, again, a CST. The operator expressed as ${\mathcal{W}}_{(a,c)}$ therefore provides a common extension of the $V_a$ and $T_c$ deformations. When $\lambda \in {\mathbf{P}}_c$, the action of ${\mathcal{W}}_{(a,c)}$ can be illustrated using the continued-fraction representation of the CST [1]. If

$${\displaystyle {\mathbf{G}}_{\lambda }\left(w\right)=\frac{1}{{\displaystyle w-{\kappa }_1-\frac{{\gamma }_1}{{\displaystyle w-{\kappa }_2-\frac{{\gamma }_2}{{\displaystyle w-{\kappa }_3-\frac{{\gamma }_3}{w-{\kappa }_4-...}}}}}}},}$$

then the CST of ${\mathcal{W}}_{(a,c)}\left(\lambda \right)$ becomes

$${\displaystyle {\mathbf{G}}_{{\mathcal{W}}_{(a,c)}\left(\lambda \right)}\left(w\right)=\frac{1}{{\displaystyle w+a{\mathbf{r}}_2\left(\lambda \right)-{\kappa }_1-\frac{{\gamma }_1}{{\displaystyle w-c-{\kappa }_2-\frac{{\gamma }_2}{{\displaystyle w-c-{\kappa }_3-\frac{{\gamma }_3}{w-c-{\kappa }_4-...}}}}}}}.}$$

This representation shows that parameters a and c act at different levels of the fraction expansion: parameter a modifies the top coefficient through the second free cumulant, while c produces a uniform translation in the deeper layers.

A natural and fundamental question arising in the study of deformations of probability measures is whether important classes of distributions preserve their structural properties under the ${\mathcal{W}}_{(a,c)}$ deformation. Addressing this question is particularly relevant in the context of CSK families, where stability under transformations often reveals deep algebraic and probabilistic structures. In this work, we prove that the Free Meixner family (FMF), one of the most prominent classes of distributions in free probability, remains invariant under the ${\mathcal{W}}_{(a,c)}$ operator. This result constitutes one of the main contributions of the paper, as it extends and unifies several previously known stability results associated with the $V_a$ and $T_c$ deformations. By embedding these two deformations into the single framework provided by the ${\mathcal{W}}_{(a,c)}$ deformation, we obtain a more comprehensive understanding of how these operators act on CSK families. This invariance property highlights the remarkable structural robustness of the FMF within the concept of CSK families. In particular, it shows that the fundamental analytic and probabilistic tools associated with quadratic VFs and free additive convolution remain valid after the deformation is applied. Consequently, the results obtained in this paper provide a unified perspective that facilitates the study of deformed convolution structures and their associated distribution families. Furthermore, the proposed framework sheds new light on the role of canonical distributions in free probability. As an additional contribution, by carefully analyzing the action of the ${\mathcal{W}}_{(a,c)}$ operator on CSK families, we derive a new characterization of the semicircle law. We prove that if the CSK family generated by a measure ($\lambda$) remains invariant under the deformation (${\mathcal{W}}_{(a,c)}$), then $\lambda$ must be, up to an affine transformation, the semicircle law. Conversely, affine images of the semicircle law, indeed, satisfy this invariance property. From a symmetry perspective, this result identifies the semicircle law as the unique fixed point, at the level of CSK families, of the ${\mathcal{W}}_{(a,c)}$ deformation. This characterization further emphasizes the distinguished and central role played by the semicircle distribution in the structure of free probability theory and in the geometry of CSK families.

The remainder of this paper is structured as follows. Section 2 reviews the fundamental concepts related to CSK families. Section 3 studies the ${\mathcal{W}}_{(a,c)}$ deformation and derives the VF associated with the deformed measure; we also prove the stability of the FMF under a ${\mathcal{W}}_{(a,c)}$ operator. Section 4 establishes a new structural property of the semicircle distribution.

2. Preliminaries

Inspired by the classical theory of natural exponential families (NEFs), the framework of CSK families has recently emerged as a valuable approach in noncommutative probability. In the classical setting, NEFs provide a coherent statistical methodology for describing probability distributions via moment–generating functions and VFs, and they play a fundamental role in statistical theory and information geometry (see [9,10,11,12]). In free probability, CSK families arise by substituting the exponential kernel for the CSK, i.e., ${(1-\theta l)}^{-1}$, thereby adapting the NEF philosophy to a noncommutative context. This kernel-based construction is naturally aligned with the analytic techniques of free probability, including subordination theory and the CST, much as the Laplace transform underpins the classical NEF framework. As a result, CSK families provide an effective setting for investigating free analogs of familiar statistical concepts, with potential implications for free harmonic analysis, operator theory and random matrix models. A systematic analytic treatment of CSK families was initiated in [13,14], where the authors developed the foundational tools that support statistical interpretations in the free setting. This line of research was later extended in [15] to cover measures whose support is constrained on one side—for instance, measures that are bounded from above, thereby further broadening the scope and applicability of the theory.

Let ${\mathbf{P}}_{ba}$ represent the class of non-degenerate probability measures whose support has an upper (one-sided) boundary. For any $\lambda \in {\mathbf{P}}_{ba}$, the transform expressed as

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

is well defined for all $0\le \theta <{\theta }_{+}^{\lambda }$ where $1/{\theta }_{+}^{\lambda }=max\{0,sup\mathrm{supp}\left(\lambda \right)\}.$ Following [15], the one-sided CSK family induced by $\lambda$ is defined by

$${\mathcal{K}}_{+}\left(\lambda \right)=\left\{{\mathbf{P}}_{\theta }^{\lambda }\left(dl\right):={\displaystyle \frac{\lambda \left(dl\right)}{{\mathbb{M}}_{\lambda }\left(\theta \right)(1-\theta l)}};\theta \in (0,{\theta }_{+}^{\lambda })\right\}.$$

This family consists of probability measures obtained by analytically deforming $\lambda$ through the $\theta$ parameter, which controls the degree of tilting of the original measure.

The mean function associated with this family, i.e., ${\mathbf{K}}_{\lambda }\left(\theta \right)=\int l{\mathbf{P}}_{\theta }^{\lambda }\left(dl\right),$ is increasing strictly on $(0,{\theta }_{+}^{\lambda })$. The corresponding image interval, i.e., $(m_1^{\lambda },m_{+}^{\lambda })={\mathbf{K}}_{\lambda }\left((0,{\theta }_{+}^{\lambda })\right),$ is called the mean domain of ${\mathcal{K}}_{+}\left(\lambda \right)$. The function expressed as ${\psi }_{\lambda }(·)$, defined as the reciprocal of ${\mathbf{K}}_{\lambda }(·)$, allows for a mean parametrization of the CSK family: For each $m\in (m_1^{\lambda },m_{+}^{\lambda })$,

$${\mathbf{Q}}_m^{\lambda }\left(dl\right)={\mathbf{P}}_{{\psi }_{\lambda }\left(m\right)}^{\lambda }\left(dl\right),{\mathcal{K}}_{+}\left(\lambda \right)=\{{\mathbf{Q}}_m^{\lambda }:m\in (m_1^{\lambda },m_{+}^{\lambda })\}.$$

Letting ${\mathbf{B}}_{\lambda }=1/{\theta }_{+}^{\lambda }$, we know from [15] that $m_1^{\lambda }={lim}_{\theta \to 0^{+}}{\mathbf{K}}_{\lambda }\left(\theta \right)$, and $m_{+}^{\lambda }={\mathbf{B}}_{\lambda }-{lim}_{w\to {\mathbf{B}}_{\lambda }^{+}}1/{\mathbf{G}}_{\lambda }\left(w\right).$

When the support of $\lambda$ is bounded from below, an analogous construction yields the lower one-sided CSK family, denoted by ${\mathcal{K}}_{-}\left(\lambda \right)$. In this case, the parameter domain becomes $\theta \in ({\theta }_{-}^{\lambda },0)$, where ${\theta }_{-}^{\lambda }$ equals either $1/{\mathbf{A}}_{\lambda }$ or $-\infty$, where ${\mathbf{A}}_{\lambda }=min\{0,infsupp\left(\lambda \right)\}$. The mean domain is then $(m_{-}^{\lambda },m_1^{\lambda })$ with $m_{-}^{\lambda }={\mathbf{A}}_{\lambda }-1/{\mathbf{G}}_{\lambda }\left({\mathbf{A}}_{\lambda }\right).$ If $\lambda$ has a two-sided bounded support (i.e., $\lambda \in {\mathbf{P}}_c$), the corresponding two-sided CSK family is defined by $\mathcal{K}\left(\lambda \right)={\mathcal{K}}_{-}\left(\lambda \right)\cup \left\{\lambda \right\}\cup {\mathcal{K}}_{+}\left(\lambda \right),$ with $\theta \in ({\theta }_{-}^{\lambda },{\theta }_{+}^{\lambda })$.

For $\lambda \in {\mathbf{P}}_{ba}$, the VF introduced in [13] is defined by

$$m\mapsto {\mathbf{V}}_{\lambda }\left(m\right)=\int {(l-m)}^2{\mathbf{Q}}_m^{\lambda }\left(dl\right).$$

It is a fundamental ingredient in the study of CSK families, as it encodes how the variance depends on the mean in a way that mirrors the role played by VFs in the theory of NEFs. This relationship offers direct insight into the intrinsic organization of CSK families and allows one to analyze their behavior under various analytic operations. In particular, the VF is instrumental in detecting stability properties and in singling out important distributions, including the free Meixner laws. Beyond these specific applications, the VF serves as a unifying analytic object that connects moments, transform-based representations, and convolution structures within free probability. Bringing these elements together in a single functional framework facilitates both structural investigations and comparative studies between free and classical probabilistic settings.

If $\lambda \in {\mathbf{P}}_{ba}$ does not possess a finite mean, then all measures in ${\mathcal{K}}_{+}\left(\lambda \right)$ have infinite variance. In this situation, the authors of [15] introduced the pseudo-variance function (PVF), i.e., ${\mathbb{V}}_{\lambda }(·)$, as a suitable extension of the VF, as defined by

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

When $m_1^{\lambda }=\int l\lambda \left(dl\right)$ is finite, both the VF and PVF exist and are related through

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

(4)

This relationship reveals that the PVF generalizes the concept of the VF to a broader setting, preserving its analytical and structural significance, even when the mean of the generating measure is not finite.

The following remark brings this section to a close and consolidates the key elements established thus far. It assembles the analytical observations, structural properties, and intermediate conclusions that will be essential in the proofs of the main results presented in this article. By summarizing these components in a unified manner, the remark provides a clear reference point for the developments that follow.

Remark 1.

Let $\lambda \in {\mathbf{P}}_{ba}$.

 (i) 

As shown in [15], PVF ${\mathbb{V}}_{\lambda }(·)$ uniquely characterizes λ: If we set $\omega =\omega \left(m\right)=m+{\displaystyle \frac{{\mathbb{V}}_{\lambda }\left(m\right)}{m}},$ then

$${\mathbf{G}}_{\lambda }\left(\omega \right)={\displaystyle \frac{m}{{\mathbb{V}}_{\lambda }\left(m\right)}}.$$

(5)

In addition, if $m_1^{\lambda }$ is finite, then

$${\mathbf{G}}_{\lambda }\left(\omega \right)={\displaystyle \frac{m-m_1^{\lambda }}{{\mathbf{V}}_{\lambda }\left(m\right)}}.$$

(6)

Consequently, VF ${\mathbf{V}}_{\lambda }(·)$, together with the mean $m_1^{\lambda }$, uniquely specifies λ.

 (ii) 

Let $\phi \left(\lambda \right)$ denote the image of the λ measure under the mapping expressed as $\phi :l\mapsto \vartheta l+\beta$ with $\vartheta \ne 0$ and $\beta \in \mathbb{R}$. Then, for m sufficiently close to $m_1^{\phi \left(\lambda \right)}=\phi \left(m_1^{\lambda }\right)=\vartheta m_1^{\lambda }+\beta$, the PVF is [15]

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

If the VF exists, then

$${\mathbf{V}}_{\phi \left(\lambda \right)}\left(m\right)={\vartheta }^2{\mathbf{V}}_{\lambda }\left({\displaystyle \frac{m-\beta }{\upsilon }}\right).$$

(7)

 (iii) 

Consider the $T_c$ deformation of λ given by (2) for $c\in \mathbb{R}$. For $m>m_1^{T_a\left(\lambda \right)}=m_1^{\lambda }$, (m close enough to $m_1^{T_a\left(\lambda \right)}$) [16],

$${\mathbb{V}}_{T_c\left(\lambda \right)}\left(m\right)=cm+{\mathbb{V}}_{\lambda }\left(m\right).$$

(8)

Furthermore, $\mathrm{Var}\left(T_c\left(\lambda \right)\right)=\mathrm{Var}\left(\lambda \right)$.

 (iv) 

For $\lambda \in {\mathbf{P}}_{ba}$ with finite variance, consider the $V_a$ deformation of λ given by (1) for $a\in \mathbb{R}$. For $m>m_1^{V_a\left(\lambda \right)}=m_1^{\lambda }-a{\mathbf{r}}_2\left(\lambda \right)$, (m close enough to $m_1^{V_a\left(\lambda \right)}$) the PVF is [17]

$${\mathbb{V}}_{V_a\left(\lambda \right)}\left(m\right)={\displaystyle \frac{m}{m+a{\mathbf{r}}_2\left(\lambda \right)}}{\mathbb{V}}_{\lambda }(m+a{\mathbf{r}}_2\left(\lambda \right))+ma{\mathbf{r}}_2\left(\lambda \right).$$

(9)

In addition, $\mathrm{Var}\left(V_a\left(\lambda \right)\right)=\mathrm{Var}\left(\lambda \right)$.

 (v) 

The free cumulant transform (${\mathcal{R}}_{\lambda }(·)$) of λ is defined by [18,19]

$${\mathcal{R}}_{\lambda }\left({\mathbf{G}}_{\lambda }\left(w\right)\right)=-1/{\mathbf{G}}_{\lambda }\left(w\right)+w,wcloseto\infty .$$

According to [20], if the variance of λ is finite, the $\mathcal{R}$ transform admits the following expansion in a neighborhood of 0:

$${\mathcal{R}}_{\lambda }\left(\eta \right)=m_1^{\lambda }+\mathrm{Var}\left(\lambda \right)\eta +\eta \epsilon \left(\eta \right),\epsilon \left(\eta \right)\stackrel{\eta \to 0}{\to }0.$$

(10)

3. Invariance of the Free Meixner Family Under ${\mathcal{W}}_{(\mathbf{a},\mathbf{c})}$ Operator

We derive a central result describing the transformation of the VF when the underlying measure is acted upon by the ${\mathcal{W}}_{(a,c)}$ deformation. This result plays a pivotal role in the paper, as it underpins the analytical framework and supports the main theorems presented in the subsequent sections.

Proposition 1.

Let $\lambda \in {\mathbf{P}}_{ba}$ be a probability measure with finite variance, and let ${\mathcal{W}}_{(a,c)}$ denote the deformation defined in (3), where a, $c\in \mathbb{R}$. Then, for $m>m_1^{{\mathcal{W}}_{(a,c)}\left(\lambda \right)}=m_1^{\lambda }-a{\mathbf{r}}_2\left(\lambda \right)$, (m close enough to $m_1^{{\mathcal{W}}_{(a,c)}\left(\lambda \right)}$), the PVF is

$${\mathbb{V}}_{{\mathcal{W}}_{(a,c)}\left(\lambda \right)}\left(m\right)={\displaystyle \frac{m}{m+a{\mathbf{r}}_2\left(\lambda \right)}}{\mathbb{V}}_{\lambda }(m+a{\mathbf{r}}_2\left(\lambda \right))+(a{\mathbf{r}}_2\left(\lambda \right)+c)m,$$

(11)

and the VF is,

$${\mathbf{V}}_{{\mathcal{W}}_{(a,c)}\left(\lambda \right)}\left(m\right)={\mathbf{V}}_{\lambda }(m+a{\mathbf{r}}_2\left(\lambda \right))+(a{\mathbf{r}}_2\left(\lambda \right)+c)(m+a{\mathbf{r}}_2\left(\lambda \right)-m_1^{\lambda }).$$

(12)

Furthermore, $\mathrm{Var}\left({\mathcal{W}}_{(a,c)}\left(\lambda \right)\right)=\mathrm{Var}\left(\lambda \right)$.

Proof. 

We begin by noting that

$${\mathcal{W}}_{(a,c)}\left(\lambda \right)=V_a\left(T_c\left(\lambda \right)\right)=T_c\left(V_a\left(\lambda \right)\right).$$

(13)

Indeed, based on (1) and (2), one sees that

$${\displaystyle \frac{1}{{\mathbf{G}}_{T_c\left(V_a\left(\lambda \right)\right)}\left(w\right)}}={\displaystyle \frac{1}{{\mathbf{G}}_{V_a\left(\lambda \right)}(w-c)}}+c={\displaystyle \frac{1}{{\mathbf{G}}_{\lambda }(w-c)}}+a{\mathbf{r}}_2\left(\lambda \right)+c={\displaystyle \frac{1}{{\mathbf{G}}_{{\mathcal{W}}_{(a,c)}\left(\lambda \right)}\left(w\right)}},$$

and

$${\displaystyle \frac{1}{{\mathbf{G}}_{V_a\left(T_c\left(\lambda \right)\right)}\left(w\right)}}={\displaystyle \frac{1}{{\mathbf{G}}_{T_c\left(\lambda \right)}\left(w\right)}}+a{\mathbf{r}}_2\left(\lambda \right)={\displaystyle \frac{1}{{\mathbf{G}}_{\lambda }(w-c)}}+c+a{\mathbf{r}}_2\left(\lambda \right)={\displaystyle \frac{1}{{\mathbf{G}}_{{\mathcal{W}}_{(a,c)}\left(\lambda \right)}\left(w\right)}}.$$

Thus,

$${\mathbf{G}}_{T_c\left(V_a\left(\lambda \right)\right)}\left(w\right)={\mathbf{G}}_{{\mathcal{W}}_{(a,c)}\left(\lambda \right)}\left(w\right)={\mathbf{G}}_{V_a\left(T_c\left(\lambda \right)\right)}\left(w\right).$$

(14)

Then, combining (14) with the Stieltjes inversion formula yields (13), thereby completing the argument. Next, for m in a neighborhood of $m_1^{{\mathcal{W}}_{(a,c)}\left(\lambda \right)}=m_1^{T_c\left(V_a\left(\lambda \right)\right)}=m_1^{V_a\left(\lambda \right)}=m_1^{\lambda }-a{\mathbf{r}}_2\left(\lambda \right)$, we have

$$\begin{array}{ccc}\hfill {\mathbb{V}}_{{\mathcal{W}}_{(a,c)}\left(\lambda \right)}\left(m\right)& \stackrel{\left(13\right)}{=}& {\mathbb{V}}_{T_c\left(V_a\left(\lambda \right)\right)}\left(m\right)\hfill \\ & \stackrel{\left(8\right)}{=}& {\mathbb{V}}_{V_a\left(\lambda \right)}\left(m\right)+cm\hfill \\ & \stackrel{\left(9\right)}{=}& {\displaystyle \frac{m}{m+a{\mathbf{r}}_2\left(\lambda \right)}}{\mathbb{V}}_{\lambda }(m+a{\mathbf{r}}_2\left(\lambda \right))+(a{\mathbf{r}}_2\left(\lambda \right)+c)m.\hfill \end{array}$$

This completes the proof of (11). Furthermore, from (11) and (4), we obtain

$${\displaystyle \frac{m}{m-(m_1^{\lambda }-a{\mathbf{r}}_2\left(\lambda \right))}}{\mathbf{V}}_{{\mathcal{W}}_{(a,c)}\left(\lambda \right)}\left(m\right)={\displaystyle \frac{m}{m+a{\mathbf{r}}_2\left(\lambda \right)}}{\displaystyle \frac{m+a{\mathbf{r}}_2\left(\lambda \right)}{m+a{\mathbf{r}}_2\left(\lambda \right)-m_1^{\lambda }}}{\mathbf{V}}_{\lambda }(m+a{\mathbf{r}}_2\left(\lambda \right))+(a{\mathbf{r}}_2\left(\lambda \right)+c)m,$$

that is,

$$\begin{array}{ccc}\hfill {\mathbf{V}}_{{\mathcal{W}}_{(a,c)}\left(\lambda \right)}\left(m\right)& =& {\displaystyle \frac{m-(m_1^{\lambda }-a{\mathbf{r}}_2\left(\lambda \right))}{m}}\times {\displaystyle \frac{m}{m+a{\mathbf{r}}_2\left(\lambda \right)}}\times {\displaystyle \frac{m+a{\mathbf{r}}_2\left(\lambda \right)}{m+a{\mathbf{r}}_2\left(\lambda \right)-m_1^{\lambda }}}\times {\mathbf{V}}_{\lambda }(m+a{\mathbf{r}}_2\left(\lambda \right))\hfill \\ & +& {\displaystyle \frac{m-(m_1^{\lambda }-a{\mathbf{r}}_2\left(\lambda \right))}{m}}\times (a{\mathbf{r}}_2\left(\lambda \right)+c)m.\hfill \\ & =& {\mathbf{V}}_{\lambda }(m+a{\mathbf{r}}_2\left(\lambda \right))+(a{\mathbf{r}}_2\left(\lambda \right)+c)(m+a{\mathbf{r}}_2\left(\lambda \right)-m_1^{\lambda }).\hfill \end{array}$$

This establishes (12). Moreover,

$$Var\left({\mathcal{W}}_{(a,c)}\left(\lambda \right)\right)=Var\left(T_c\left(V_a\left(\lambda \right)\right)\right)=Var\left(V_a\left(\lambda \right)\right)=Var\left(\lambda \right).$$

□

The stability of the FMF under measure deformations is a key feature in free probability theory. This property expresses the robustness of the FMF with respect to fundamental operations, including free convolution and the related analytic deformations, and sheds light on the relationships between free cumulants, convolution semigroups, and Voiculescu transforms. Owing to its rich analytic and algebraic structure, as well as its ability to encompass a broad range of significant distributions, the invariance of the FMF is essential for applications such as classification problems, modeling, and asymptotic analysis, particularly in random matrix theory. An important illustration arises from the class of quadratic CSK families whose VFs are of the following form:

$${\mathbf{V}}_{\lambda }\left(m\right)=1+\tau m+\varsigma m^2,\tau \in \mathbb{R},\varsigma \ge -1,m_1^{\lambda }=0.$$

(15)

These families were completely characterized in [13], and the associated generating measures belong to the FMF. They admit the following explicit representation:

$$\lambda \left(dl\right)={\displaystyle \frac{\sqrt{4(1+\varsigma )-{(l-\tau )}^2}}{2\pi (\varsigma l^2+\tau l+1)}}{1}_{(-2\sqrt{1+\varsigma }+\tau ,+2\sqrt{1+\varsigma }+\tau )}\left(l\right)dl+p_1{\delta }_{l_1}+p_2{\delta }_{l_2}.$$

(16)

This formula highlights the mixed nature of these measures, combining absolutely continuous and discrete components, and reflects the structural complexity of the FMF. More precisely,

(i)

If $\varsigma =0,{\tau }^2>1$, then $p_1=1-1/{\tau }^2,l_1=-1/\tau ,p_2=0$.

(ii)

If $\varsigma >0$ and ${\tau }^2>4\varsigma$, then $p_1=max\{0,1-{\displaystyle \frac{\left|\tau \right|-\sqrt{{\tau }^2-4\varsigma }}{2\varsigma \sqrt{{\tau }^2-4\varsigma }}}\},p_2=0$, and $l_1=\pm {\displaystyle \frac{\left|\tau \right|-\sqrt{{\tau }^2-4\varsigma }}{2\varsigma }}$ with the sign opposite to the sign of $\tau$.

(iii)

If $\varsigma \in [-1,0)$, then

$$l_{1,2}={\displaystyle \frac{-\tau \pm \sqrt{{\tau }^2-4\varsigma }}{2\varsigma }},p_{1,2}=1+{\displaystyle \frac{\sqrt{{\tau }^2-4\varsigma }\mp \tau }{2\varsigma \sqrt{{\tau }^2-4\varsigma }}}.$$

This framework encompasses several prominent distributions. Up to scaling and free convolution, the $\lambda$ measure corresponds to:

(i)

The semicircle ($\mathbb{SC}$) type distribution if $\tau =\varsigma =0$;

(ii)

The Marchenko–Pastur ($\mathbb{MP}$) type distribution if $\varsigma =0$ and $\tau \ne 0$;

(iii)

The free Pascal ($\mathbb{FP}$) type distribution if $\varsigma >0$ and ${\tau }^2>4\varsigma$;

(iv)

The free Gamma ($\mathbb{FG}$) type distribution if $\varsigma >0$ and ${\tau }^2=4\varsigma$;

(v)

The free analog of hyperbolic ($\mathbb{FH}$) type distribution if $\varsigma >0$ and ${\tau }^2<4\varsigma$;

(vi)

The free binomial ($\mathbb{FB}$) type distribution if $-1\le \varsigma <0$.

Within this setting, we prove that the deformation (${\mathcal{W}}_{(a,c)}$) preserves the FMF, thereby extending its invariance properties under a broader class of analytic deformations.

Theorem 1.

If $\lambda \in \mathrm{FMF}$, then ∀ a, $c\in \mathbb{R}$, and one has ${\mathbf{T}}_{a{\mathbf{r}}_2\left(\lambda \right)}\left({\mathcal{W}}_{(a,c)}\left(\lambda \right)\right)\in \mathrm{FMF}$, where ${\mathbf{T}}_{\beta }\left(l\right)=l+\beta$ with $\beta \in \mathbb{R}$.

Proof. 

For $\lambda \in \mathrm{FMF}$, one has ${\mathbf{r}}_2\left(\lambda \right)=1$, and the associated VF can be expressed, for m in a neighborhood of $m_1^{\lambda }=0$, as

$${\mathbf{V}}_{\lambda }\left(m\right)=1+{\tau }^{\prime }m+{\varsigma }^{\prime }{m}^2,{\tau }^{\prime }\in \mathbb{R},{\varsigma }^{\prime }\ge -1.$$

(17)

For m close enough to $m_1^{{\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(\lambda \right)\right)}=0$, one gets

$$\begin{array}{ccc}\hfill {\mathbf{V}}_{{\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(\lambda \right)\right)}\left(m\right)& \stackrel{\left(7\right)}{=}& {\mathbf{V}}_{{\mathcal{W}}_{(a,c)}\left(\lambda \right)}(m-a)\hfill \\ & \stackrel{\left(12\right)}{=}& {\mathbf{V}}_{\lambda }\left(m\right)+(a+c)m\hfill \\ & \stackrel{\left(17\right)}{=}& 1+\left({\tau }^{\prime }+a+c\right)m+{\varsigma }^{\prime }{m}^2,\hfill \end{array}$$

(18)

which is a VF of the kind specified in (15). Then, ${\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(\lambda \right)\right)\in \mathrm{FMF}$. □

Theorem 1 shows that the FMF remains invariant under the action of the ${\mathcal{W}}_{(a,c)}$ deformation. This stability highlights the structural coherence of the FMF within the framework of noncommutative probability and confirms its resilience under a broad class of deformations. Such a feature further emphasizes the central role of this family in capturing invariant phenomena in free probability, and it motivates a more detailed analytical and structural investigation. We now turn to a collection of examples that illustrate the scope and relevance of Theorem 1.

Corollary 1.

Let $sb\left(dl\right)={\displaystyle \frac{1}{2}}{\delta }_{-1}+{\displaystyle \frac{1}{2}}{\delta }_1$. Then, the ${\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(sb\right)\right)$ measure is a $\mathbb{FB}$-type law (16) with parameters of $\varsigma =-1$ and $\tau =c+a$.

Proof. 

For the symmetric Bernoulli law ($sb$), the associated VF is given by (17) with parameters of ${\varsigma }^{\prime }=-1$ and ${\tau }^{\prime }=0$. Applying (18), we obtain

$${\mathbf{V}}_{{\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(sb\right)\right)}\left(m\right)=1+(a+c)m-m^2,m\mathrm{in}\mathrm{a}\mathrm{neighborhood} \mathrm{of} 0.$$

Comparing this expression with the quadratic form (15) yields the desired conclusion, in view of (6). □

The $\mathbb{SC}$ law occupies a fundamental role in free probability, making it important to analyze its behavior under the ${\mathcal{W}}_{(a,c)}$ operator.

Corollary 2.

Take the $\mathbb{SC}$ law, i.e.,

$${\displaystyle sc\left(dl\right)=\frac{1}{2\pi }\sqrt{4-l^2}{1}_{(-2,2)}\left(l\right)dl.}$$

(19)

Then, the ${\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(sc\right)\right)$ measure is a

 (i) 

$\mathbb{SC}$-type law if $c=-a$;

 (ii) 

$\mathbb{MP}$-type law with $\varsigma =0$ and $\tau =c+a$ if $c\ne -a$.

Proof. 

For the $\mathbb{SC}$ law ($sc$), the associated VF is given by (17) with parameters of ${\varsigma }^{\prime }={\tau }^{\prime }=0$. Applying (18) yields

$${\mathbf{V}}_{{\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(sc\right)\right)}\left(m\right)=1+(a+c)m,m\mathrm{in}\mathrm{a}\mathrm{neighborhood}\mathrm{of} 0.$$

(20)

By comparing (20) with (15), we obtain the following:

(i)

If $c=-a$, then ${\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(sc\right)\right)=sc$.

(ii)

If $c\ne -a$, then ${\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(sc\right)\right)$ is an $\mathbb{MP}$ distribution (16) with $\varsigma =0$ and $\tau =c+a$.

□

The $\mathbb{MP}$ law is a central object in noncommutative probability, making it important to examine its behavior under the ${\mathcal{W}}_{(a,c)}$ operator.

Corollary 3.

For ${\varsigma }^{\prime }=0$ and ${\tau }^{\prime }\ne 0$, take the $\mathbb{MP}$ law, i.e.,

$$m{p}_{{\tau }^{\prime }}\left(dl\right)={\displaystyle \frac{\sqrt{4-{(l-{\tau }^{\prime })}^2}}{2\pi ({\tau }^{\prime }l+1)}}{1}_{(-2+{\tau }^{\prime },2+{\tau }^{\prime })}\left(l\right)dl+{(-1/{\tau }^{\prime 2}+1)}^{+}{\delta }_{-1/{\tau }^{\prime }}.$$

Then, the ${\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(m{p}_{{\tau }^{\prime }}\right)\right)$ measure is a

 (i) 

$\mathbb{SC}$-type law where $\tau =a+c+{\tau }^{\prime }=0$ and $\varsigma =0={\varsigma }^{\prime }$ if $a+c+{\tau }^{\prime }=0$;

 (ii) 

$\mathbb{MP}$-type law where $\tau =a+c+{\tau }^{\prime }$ and $\varsigma =0={\varsigma }^{\prime }$ if $a+c+{\tau }^{\prime }\ne 0$.

Proof. 

For the $\mathbb{MP}$ law ($m{p}_{{\tau }^{\prime }}$), the associated VF is given by (17) with parameters of ${\varsigma }^{\prime }=0$ and ${\tau }^{\prime }\ne 0$. Applying (18) yields

$${\mathbf{V}}_{{\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(m{p}_{{\tau }^{\prime }}\right)\right)}\left(m\right)=1+({\tau }^{\prime }+a+c)m,m\mathrm{in}\mathrm{a}\mathrm{neighborhood}\mathrm{of} 0.$$

(21)

By comparing (21) and (15), we obtain the following:

(i)

If ${\tau }^{\prime }+a+c=0$, then ${\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(m{p}_{{\tau }^{\prime }}\right)\right)$ is an $\mathbb{SC}$ distribution (16) with $\tau =a+c+{\tau }^{\prime }=0$ and $\varsigma =0$.

(ii)

If $a+c+{\tau }^{\prime }\ne 0$, then ${\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(m{p}_{{\tau }^{\prime }}\right)\right)$ is an $\mathbb{MP}$ distribution (16) with $\tau ={\tau }^{\prime }+a+c$ and $\varsigma =0$.

□

It is likewise essential to examine the behavior of the $\mathbb{FG}$ law under the ${\mathcal{W}}_{(a,c)}$ operator.

Corollary 4.

For ${\varsigma }^{\prime }={\tau }^{\prime 2}/4$ with ${\tau }^{\prime }\ne 0$, take the $\mathbb{FG}$ law, i.e.,

$$f{g}_{({\tau }^{\prime },{\varsigma }^{\prime })}\left(dl\right)={\displaystyle \frac{\sqrt{4(1+{\varsigma }^{\prime })-{(l-{\tau }^{\prime })}^2}}{2\pi ({\varsigma }^{\prime }{l}^2+{\tau }^{\prime }l+1)}}{1}_{(-2\sqrt{1+{\varsigma }^{\prime }}+{\tau }^{\prime },2\sqrt{1+{\varsigma }^{\prime }}+{\tau }^{\prime })}\left(l\right)dl.$$

Then, ${\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(f{g}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)\right)$ is a

 (i) 

$\mathbb{FG}$ distribution with $\tau ={\tau }^{\prime }$ and $\varsigma ={\tau }^{\prime 2}/4={\varsigma }^{\prime }$ if $a=-c$;

 (ii) 

$\mathbb{FP}$ distribution with $\tau =a+c+{\tau }^{\prime }$ and $\varsigma ={\varsigma }^{\prime }={\tau }^{\prime 2}/4$ if ${(a+c+{\tau }^{\prime })}^2>{\tau }^{\prime 2}$.

 (iii) 

$\mathbb{FH}$ distribution with $\tau =a+c+{\tau }^{\prime }$ and $\varsigma ={\varsigma }^{\prime }={\tau }^{\prime 2}/4$ if ${(a+c+{\tau }^{\prime })}^2<{\tau }^{\prime 2}$.

Proof. 

For the $\mathbb{FG}$ distribution ($f{g}_{({\tau }^{\prime },{\varsigma }^{\prime })}$), the corresponding VF is given by (17) for ${\varsigma }^{\prime }={\tau }^{\prime 2}/4$ with ${\tau }^{\prime }\ne 0$. Relation (18) gives

$${\mathbf{V}}_{{\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(f{g}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)\right)}\left(m\right)=1+({\tau }^{\prime }+a+c)m+{\tau }^{\prime 2}{m}^2/4,m\mathrm{sufficiently}\mathrm{close}\mathrm{to} 0.$$

(22)

By matching (22) with the form in (15), we get the following:

(i)

If ${({\tau }^{\prime }+a+c)}^2={\tau }^{\prime 2}$ (i.e., $a=-c$), then ${\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(f{g}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)\right)=f{g}_{({\tau }^{\prime },{\varsigma }^{\prime })}$.

(ii)

If ${({\tau }^{\prime }+a+c)}^2>{\tau }^{\prime 2}$, then ${\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(f{g}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)\right)$ is an $\mathbb{FP}$-type law (16) with $\tau =a+c+{\tau }^{\prime }$ and $\varsigma ={\varsigma }^{\prime }={\tau }^{\prime 2}/4$.

(iii)

If ${(a+c+{\tau }^{\prime })}^2<{\tau }^{\prime 2}$, then ${\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(f{g}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)\right)$ is an $\mathbb{FH}$-type law (16) with $\tau =a+c+{\tau }^{\prime }$ and $\varsigma ={\varsigma }^{\prime }={\tau }^{\prime 2}/4$.

□

Finally, we examine how the $\mathbb{FH}$ distribution behaves under the action of the ${\mathcal{W}}_{(a,c)}$ operator.

Corollary 5.

For ${\varsigma }^{\prime }>0$ and ${\tau }^{\prime 2}<4{\varsigma }^{\prime }$, take the $\mathbb{FH}$ law, i.e.,

$$f{h}_{({\tau }^{\prime },{\varsigma }^{\prime })}\left(dl\right)={\displaystyle \frac{\sqrt{4(1+{\varsigma }^{\prime })-{(l-{\tau }^{\prime })}^2}}{2\pi ({\varsigma }^{\prime }{l}^2+{\tau }^{\prime }l+1)}}{1}_{(-2\sqrt{{\varsigma }^{\prime }+1}+{\tau }^{\prime },2\sqrt{{\varsigma }^{\prime }+1}+{\tau }^{\prime })}\left(l\right)dl.$$

Then, ${\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(f{h}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)\right)$ is

 (i) 

An $\mathbb{FG}$-type law (16) with $\tau =a+c+{\tau }^{\prime }$ and $\varsigma ={\varsigma }^{\prime }$ if ${(a+c+{\tau }^{\prime })}^2=4{\varsigma }^{\prime }$;

 (ii) 

An $\mathbb{FP}$-type law (16) with $\tau =a+c+{\tau }^{\prime }$ and $\varsigma ={\varsigma }^{\prime }$ if ${(a+c+{\tau }^{\prime })}^2>4{\varsigma }^{\prime }$.

 (iii) 

An $\mathbb{FH}$-type law (16) with $\tau =a+c+{\tau }^{\prime }$ and $\varsigma ={\varsigma }^{\prime }$ if ${(a+c+{\tau }^{\prime })}^2<4{\varsigma }^{\prime }$.

Proof. 

For the $\mathbb{FH}$ distribution ($f{h}_{({\tau }^{\prime },{\varsigma }^{\prime })}$), the related VF is given by (17) for ${\varsigma }^{\prime }>0$ and ${\tau }^{\prime 2}<4{\varsigma }^{\prime }$. Applying (18) yields

$${\mathbf{V}}_{{\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(f{h}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)\right)}\left(m\right)=1+({\tau }^{\prime }+a+c)m+{\varsigma }^{\prime }{m}^2,m\mathrm{in}\mathrm{a}\mathrm{neighborhood}\mathrm{of} 0.$$

(23)

By matching (23) with the form in (15), we get the following:

(i)

If ${(a+c+{\tau }^{\prime })}^2=4{\varsigma }^{\prime }$, then ${\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(f{h}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)\right)$ is an $\mathbb{FG}$-type distribution (16) with $\tau =a+c+{\tau }^{\prime }$ and $\varsigma ={\varsigma }^{\prime }$.

(ii)

If ${(a+c+{\tau }^{\prime })}^2>4{\varsigma }^{\prime }$, then ${\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(f{h}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)\right)$ is an $\mathbb{FP}$-type distribution (16) with $\tau =a+c+{\tau }^{\prime }$ and $\varsigma ={\varsigma }^{\prime }$.

(iii)

If ${(a+c+{\tau }^{\prime })}^2<4{\varsigma }^{\prime }$, then ${\mathbf{T}}_a\left({\mathcal{W}}_{(a,c)}\left(f{h}_{({\tau }^{\prime },{\varsigma }^{\prime })}\right)\right)$ is an $\mathbb{FH}$-type distribution (16) with $\tau =a+c+{\tau }^{\prime }$ and $\varsigma ={\varsigma }^{\prime }$.

□

We examined the effects of the unified operator (${\mathcal{W}}_{(a,c)}$) using explicit analytical examples. These case studies demonstrate how classical distributions in noncommutative probability can be systematically obtained, modified, or linked within this single framework. The examples emphasize the operator’s capacity to maintain key structural features, uncover previously unnoticed relationships, and produce new families of distributions with clear probabilistic significance. Overall, this investigation highlights the role of ${\mathcal{W}}_{(a,c)}$ as a unifying tool that connects different convolution mechanisms and provides a basis for further theoretical exploration.

4. A Property of the Semicircle Law

Let $\lambda \in {\mathbf{P}}_{ba}$ be a generating measure with a finite mean. A natural question is how the corresponding CSK family (${\mathcal{K}}_{+}\left(\lambda \right)$) transforms under the action of the operator (${\mathcal{W}}_{(a,c)}$). Applying this operator to each element of the family produces the collection expressed as

$${\mathcal{W}}_{(a,c)}\left({\mathcal{K}}_{+}\left(\lambda \right)\right)=\{{\mathcal{W}}_{(a,c)}\left({\mathbf{Q}}_m^{\lambda }\right):{\mathbf{Q}}_m^{\lambda }\in {\mathcal{K}}_{+}\left(\lambda \right)\},$$

(24)

which captures the joint influence of the shift and deformation parameters encoded in ${\mathcal{W}}_{(a,c)}$. An important observation is that the exact structure of this transformed class provides a characterization property. Specifically, for any nonzero pair ($(a,c)\ne (0,0)$), the identity expressed as

$${\mathcal{W}}_{(a,c)}\left({\mathcal{K}}_{+}\left(\lambda \right)\right)={\mathcal{K}}_{+}\left({\mathbf{T}}_c\left(\lambda \right)\right)$$

(25)

is satisfied if and only if the generating measure ($\lambda$) belongs, up to an affine change of variables, to the class of $\mathbb{SC}$ distributions. This establishes the special status of the $\mathbb{SC}$ law in the setting of CSK families: it is the only distribution whose associated CSK family preserves its form under the full two-parameter transformation (${\mathcal{W}}_{(a,c)}$).

In preparation for proving the main theorem of this section (Theorem 2), we first need a supplementary result describing the behavior of the mean domain. Although technical in nature, this observation is essential for tracking how the domain of means transforms and for ensuring that the subsequent arguments are rigorously justified.

Remark 2.

Let $\lambda \in {\mathbf{P}}_{ba}$. Its first moment ($m_1^{\lambda }$) has straightforward properties: it behaves affinely under affine changes of variables and scales proportionally under free additive convolution powers. More precisely,

$$m_1^{\phi \left(\lambda \right)}=\phi \left(m_1^{\lambda }\right),and{m}_1^{{\lambda }^{⊞p}}=p{m}_1^{\lambda }$$

whenever the convolution power (${\lambda }^{⊞p}$) is defined.

In contrast, the upper bound ($m_{+}^{\lambda }$) does not admit a similarly simple transformation rule (see [21] (Example 3.9)). To overcome this limitation, one introduces an enlarged version of the mean domain that is compatible with the pseudo-variance (or variance) function. The extended upper endpoint is introduced as [21]

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

This quantity enjoys natural transformation properties. For any $p>0$,

$${\mathit{M}}_{+}^{{\lambda }^{⊞p}}=p{\mathit{M}}_{+}^{\lambda },$$

and for scaling and translation operators ${\mathbf{H}}_{\vartheta }\left(l\right)=l\vartheta$, ($\vartheta >0$) and ${\mathbf{T}}_{\beta }\left(l\right)=\beta +l$, $(\beta \in \mathbb{R})$, one has

$${\mathit{M}}_{+}^{{\mathbf{H}}_{\vartheta }\left(\lambda \right)}=\vartheta {\mathit{M}}_{+}^{\lambda }and{\mathit{M}}_{+}^{{\mathbf{T}}_{\beta }\left(\lambda \right)}={\mathit{M}}_{+}^{\lambda }+\beta .$$

If the support of λ is bounded from below, a symmetric construction yields

$${\mathit{M}}_{-}^{\lambda }=sup\left\{u<m_1^{\lambda }:{\mathbb{V}}_{\lambda }\left(u\right)/u<0\right\},$$

which satisfies analogous relations:

$${\mathit{M}}_{-}^{{\lambda }^{⊞p}}=p{\mathit{M}}_{-}^{\lambda },{\mathit{M}}_{-}^{{\mathbf{H}}_{\vartheta }\left(\lambda \right)}=\vartheta {\mathit{M}}_{-}^{\lambda }and{\mathit{M}}_{-}^{{\mathbf{T}}_{\beta }\left(\lambda \right)}={\mathit{M}}_{-}^{\lambda }+\beta .$$

Accordingly, in the sequel, the domain of means is chosen as $(m_1^{\lambda },{\mathbf{M}}_{+}^{\lambda })$ for $\lambda \in {\mathbf{P}}_{ba}$, as $({\mathbf{M}}_{-}^{\lambda },m_1^{\lambda })$ when the support is bounded from below, and as $({\mathbf{M}}_{-}^{\lambda },{\mathbf{M}}_{+}^{\lambda })$ when λ has compact support, i.e., $\lambda \in {\mathbf{P}}_c$.

We now state the main theorem of this section, which characterizes the $\mathbb{SC}$ distribution via the action of the ${\mathcal{W}}_{(a,c)}$ operator. This result synthesizes the preceding analysis and shows that the two-parameter transformation provides a precise criterion for singling out the $\mathbb{SC}$ law among CSK families.

Theorem 2.

Let $\lambda \in {\mathbf{P}}_{ba}$ with a finite mean, and consider the collection of measures defined in (24). Then, the identity (25) is satisfied for every $(a,c)\ne (0,0)$ if and only if λ is, up to an affine transformation, an $\mathbb{SC}$ distribution described in (19).

Proof. 

$(\Rightarrow )$ Assume that Equation (25) is satisfied ∀$(a,c)\ne (0,0)$. Then ∀$t\in (m_1^{\lambda },{\mathbf{M}}_{+}^{\lambda })$, there is $s\in (m_1^{{\mathbf{T}}_c\left(\lambda \right)},{\mathbf{M}}_{+}^{{\mathbf{T}}_c\left(\lambda \right)})$ so that

$${\mathcal{W}}_{(a,c)}\left({\mathbf{Q}}_t^{\lambda }\right)={\mathbf{Q}}_s^{{\mathbf{T}}_c\left(\lambda \right)},$$

(26)

that is,

$${\mathcal{R}}_{{\mathcal{W}}_{(a,c)}\left({\mathbf{Q}}_t^{\lambda }\right)}\left(\eta \right)={\mathcal{R}}_{{\mathbf{Q}}_s^{{\mathbf{T}}_c\left(\lambda \right)}}\left(\eta \right),\eta close to0.$$

(27)

Using (10), Equation (27) is

$$m_1^{{\mathcal{W}}_{(a,c)}\left({\mathbf{Q}}_t^{\lambda }\right)}+\mathrm{Var}\left({\mathcal{W}}_{(a,c)}\left({\mathbf{Q}}_t^{\lambda }\right)\right)\eta +\eta {\epsilon }_1\left(\eta \right)=m_1^{{\mathbf{Q}}_s^{{\mathbf{T}}_c\left(\lambda \right)}}+\mathrm{Var}\left({\mathbf{Q}}_s^{{\mathbf{T}}_c\left(\lambda \right)}\right)\eta +\eta {\epsilon }_2\left(\eta \right),{\epsilon }_{i=1,2}\left(\eta \right)\stackrel{\eta \to 0}{\to }0.$$

(28)

Knowing that $m_1^{{\mathcal{W}}_{(a,c)}\left({\mathbf{Q}}_t^{\lambda }\right)}=t-a{\mathbf{V}}_{\lambda }\left(t\right)$, $\mathrm{Var}\left({\mathcal{W}}_{(a,c)}\left({\mathbf{Q}}_t^{\lambda }\right)\right)=\mathrm{Var}\left({\mathbf{Q}}_t^{\lambda }\right)={\mathbf{V}}_{\lambda }\left(t\right)$, $m_1^{{\mathbf{Q}}_s^{{\mathbf{T}}_c\left(\lambda \right)}}=s$ and $\mathrm{Var}\left({\mathbf{Q}}_s^{{\mathbf{T}}_c\left(\lambda \right)}\right)={\mathbf{V}}_{{\mathbf{T}}_c\left(\lambda \right)}\left(s\right)={\mathbf{V}}_{\lambda }(s-c)$, relation (28) becomes

$$t-a{\mathbf{V}}_{\lambda }\left(t\right)+{\mathbf{V}}_{\lambda }\left(t\right)\eta +\eta {\epsilon }_1\left(\eta \right)=s+{\mathbf{V}}_{\lambda }(s-c)\eta +\eta {\epsilon }_2\left(\eta \right).$$

(29)

Relation (29) implies that $s=t-a{\mathbf{V}}_{\lambda }\left(t\right)$; then,

$${\mathbf{V}}_{\lambda }(t-a{\mathbf{V}}_{\lambda }\left(t\right)-c)={\mathbf{V}}_{\lambda }\left(t\right)\forall (a,c)\ne (0,0)and\forall t\in (m_1^{\lambda },{\mathbf{M}}_{+}^{\lambda }).$$

(30)

Because $\lambda$ is not degenerate, ${\mathbf{V}}_{\lambda }(·)$ cannot be identically zero. Hence, relation (30) implies that ${\mathbf{V}}_{\lambda }(·)$ must, in fact, be constant; this constant is denoted by $\sigma >0$ so that ${\mathbf{V}}_{\lambda }(·)=\sigma$. It follows that $\lambda$ can be expressed as an affine image of the $\mathbb{SC}$ distribution given in (19), obtained through a scaling by $\sqrt{\sigma }$ and a shift by $m_1^{\lambda }$.

$(\Leftarrow )$ Suppose that $\lambda$ is obtained from the $\mathbb{SC}$ distribution in (19) via the affine transformation ($l\mapsto \sqrt{\sigma }l+m_1^{\lambda }$). It remains to be verified that

$${\mathcal{W}}_{(a,c)}\left({\mathbf{Q}}_t^{\lambda }\right)={\mathbf{Q}}_{t-a\sigma }^{{\mathbf{T}}_c\left(\lambda \right)}.$$

(31)

It is clear that $m_1^{{\mathcal{W}}_{(a,c)}\left({\mathbf{Q}}_t^{\lambda }\right)}=t-a\sigma =m_1^{{\mathbf{Q}}_{t-a\sigma }^{{\mathbf{T}}_c\left(\lambda \right)}}$. Then, there is $\varsigma >$ so that ${\mathbb{V}}_{{\mathcal{W}}_{(a,c)}\left({\mathbf{Q}}_t^{\lambda }\right)}(·)$ and ${\mathbb{V}}_{{\mathbf{Q}}_{t-a\sigma }^{{\mathbf{T}}_c\left(\lambda \right)}}(·)$ are well defined on $(t-a\sigma ,t-a\sigma +\varsigma )$.

From [16] (Equation (2.12)), we have

$${\mathbb{V}}_{{\mathbf{Q}}_t^{\lambda }}\left(y\right)=y\left({\displaystyle \frac{\sigma }{y-t}}-t+m_1^{\lambda }\right),y\ne t,$$

(32)

and from [16] (Equation (2.18)), we have

$${\mathbb{V}}_{{\mathbf{Q}}_s^{{\mathbf{T}}_c\left(\lambda \right)}}\left(y\right)=y\left({\displaystyle \frac{\sigma }{y-s}}+c-s+m_1^{\lambda }\right)y\ne s.$$

(33)

Now, based on (32) and (11) and knowing that ${\mathbf{V}}_{\lambda }\left(t\right)=\sigma$, one sees that

$${\mathbb{V}}_{{\mathcal{W}}_{(a,c)}\left({\mathbf{Q}}_t^{\lambda }\right)}\left(y\right)=y\left({\displaystyle \frac{\sigma }{y+a\sigma -t}}-t+a\sigma +c+m_1^{\lambda }\right),y\ne t-a\sigma ,$$

(34)

Also, with $s=t-a{\mathbf{V}}_{\lambda }\left(t\right)=t-a\sigma$, it is clear from (33) that

$${\mathbb{V}}_{{\mathbf{Q}}_{t-a\sigma }^{{\mathbf{T}}_c\left(\lambda \right)}}\left(y\right)=y\left({\displaystyle \frac{\sigma }{y+a\sigma -t}}+c-t+a\sigma +m_1^{\lambda }\right)y\ne t-a\sigma .$$

(35)

Equations (34) and (35) give

$${\mathbb{V}}_{{\mathcal{W}}_{(a,c)}\left({\mathbf{Q}}_t^{\lambda }\right)}\left(y\right)={\mathbb{V}}_{{\mathbf{Q}}_{t-a\sigma }^{{\mathbf{T}}_c\left(\lambda \right)}}\left(y\right),y\in (t-a\sigma ,t-a\sigma +\varsigma ).$$

(36)

Identity (36) completes the derivation of (31) once relation (5) is invoked. □

Remark 3.

In the case of the $\mathbb{SC}$ distribution, the domain of means is unbounded on both sides, namely

$${\mathit{M}}_{-}^{sc}=-\infty and{\mathit{M}}_{+}^{sc}=+\infty .$$

Thus, the mean parameter is free to take any real value, without restriction. As a result, the functional identity (30), expressed in terms of the mean variable over $({\mathit{M}}_{-}^{\lambda },{\mathit{M}}_{+}^{\lambda })$, extends naturally to the entire real line, with no boundary issues or singular behavior. In particular, (30) is well posed in this setting, which justifies its use in subsequent arguments and ensures that all related identities remain valid, without imposing extra conditions on the parameter range.

5. Conclusions and Perspectives

Within the framework of CSK families, we have obtained an explicit and general formula for the VF of the family induced by a measure subjected to the ${\mathcal{W}}_{(a,c)}$ operator: an operator that brings together two key constructions from free probability—namely, the $T_c$- and $V_a$ transformations—in a single unified mechanism. The resulting formula provides an effective way to track how the VF is modified under this deformation and is central to our analysis. One important consequence is that the FMF remains unchanged under the action of ${\mathcal{W}}_{(a,c)}$, underscoring its intrinsic stability within this broader setting. Moreover, examining how ${\mathcal{W}}_{(a,c)}$ acts on CSK families leads to a new characterization of the $\mathbb{SC}$ law, thereby shedding further light on its distinguished position in free probability.

This study also opens the door to a number of natural and potentially fruitful research directions. One possible direction for future research concerns the interaction between the ${\mathcal{W}}_{(a,c)}$ deformation and other fundamental concepts of free probability. In particular, it would be interesting to investigate how this transformation affects quantities such as free entropy and whether it preserves structural properties that arise in random matrix models. Since many distributions belonging to the FMF appear as limiting spectral laws of random matrices, the stability of this family under the ${\mathcal{W}}_{(a,c)}$ deformation suggests that related random matrix interpretations may also exist. Exploring these potential connections could provide further insight into the probabilistic and analytical significance of the ${\mathcal{W}}_{(a,c)}$ framework and constitutes a natural topic for future investigation.

Another source of motivation comes from the $V_a$-deformed measure, which led to the definition of a deformed convolution, as indicated by $\begin{array}{|c|}\hline a\\ \hline\end{array}$ in [6]. For ${\lambda }_1$, ${\lambda }_2\in \mathbf{P}$ with finite variances, their $V_a$-deformed free convolution is given by

$${\lambda }_1\begin{array}{|c|}\hline a\\ \hline\end{array}{\lambda }_2=V_{-a}(V_a\left({\lambda }_1\right)⊞V_a\left({\lambda }_2\right)).$$

This approach points toward a general setting in which measure transformations and free convolution combine in a structured and analytically manageable way. A notable property is that the inverse of the ${\mathcal{W}}_{(a,c)}$ operator is simply

$${\mathcal{W}}_{(a,c)}^{-1}={\mathcal{W}}_{(-a,-c)},a,c\in \mathbb{R}.$$

(37)

To illustrate this, let $\lambda \in \mathbf{P}$ with finite variance and let $a_1,$$a_2,$$c_1$, $c_2\in \mathbb{R}$ such that

$${\mathcal{W}}_{(a_1,c_1)}\left({\mathcal{W}}_{(a_2,c_2)}\left(\lambda \right)\right)=\lambda .$$

Using identity (3), it follows that

$${\displaystyle \frac{1}{{\mathbf{G}}_{\lambda }\left(w\right)}}={\displaystyle \frac{1}{{\mathbf{G}}_{{\mathcal{W}}_{(a_1,c_1)}\left({\mathcal{W}}_{(a_2,c_2)}\left(\lambda \right)\right)}\left(w\right)}}={\displaystyle \frac{1}{{\mathbf{G}}_{\lambda }(w-c_1-c_2)}}+{\mathbf{r}}_2\left(\lambda \right)[a_1+a_2]+c_1+c_2.$$

This equality can occur only when $a_2=-a_1$ and $c_2=-c_1$, which verifies the inverse formula stated in (37).

Building on this framework, the concept of deformation can be generalized further. For any real parameters (a and c), we define a new deformed free convolution, denoted by ${⊞}_{{\mathcal{W}}_{(a,c)}}$, using the ${\mathcal{W}}_{(a,c)}$ operator. Specifically, for measures ${\lambda }_1,$${\lambda }_2\in \mathbf{P}$ with finite variances, set

$${\lambda }_1{⊞}_{{\mathcal{W}}_{(a,c)}}{\lambda }_2={\mathcal{W}}_{(-a,-c)}({\mathcal{W}}_{(a,c)}\left({\lambda }_1\right)⊞{\mathcal{W}}_{(a,c)}\left({\lambda }_2\right))a,c\in \mathbb{R}.$$

When $c=0$, this construction reduces to the previously introduced convolution ($\begin{array}{|c|}\hline a\\ \hline\end{array}$), demonstrating that ${⊞}_{{\mathcal{W}}_{(a,c)}}$ is a natural extension of earlier deformation schemes. This new operation opens a rich avenue for exploration, offering a flexible analytic framework that interpolates between different deformation and free probability structures. Its study may uncover new classes of limiting laws, provide deeper understanding of VF geometry, and potentially reveal connections to operator–algebraic methods and random matrix theory.