On a New Extension of the t-Transformation of Probability Measures

Abstract

This paper establishes a comprehensive analytical framework for a new transformation of probability measures, denoted by ${\mathcal{T}}_{(a,t)}$, which unifies the classical t- and $T_a$-transformations in free probability. We derive the functional equation characterizing ${\mathcal{T}}_{(a,t)}$ through the Cauchy–Stieltjes transform and explicitly show how it specializes to known deformations when $a=0$ or $t=1$. Within the setting of Cauchy-Stieltjes kernel families, we prove structural symmetry and invariance properties of the transformation, demonstrating in particular that both the free Meixner family and the free analog of the Letac-Mora class remain invariant under ${\mathcal{T}}_{(a,t)}$. Furthermore, we obtain several new limiting theorems that uncover symmetric relationships among fundamental free distributions, including the semicircular, Marchenko–Pastur, and free binomial laws.

1. Introduction

In the framework of free probability theory, the investigation of measure deformations has emerged as a powerful analytical tool for elucidating the structural relationships between various notions of convolution and their associated analytic representations. These deformations establish symmetric correspondences among different probabilistic models, illustrating how probability measures transform under distinct convolution operations. Two prominent examples are the t-deformation [1,2] and the $T_a$-deformation [3,4]. The t-deformation provides a continuous interpolation between classical, free, and Boolean convolutions, whereas the $T_a$-deformation functions as a homomorphic mapping connecting free, monotone, and Boolean convolutions. Each transformation captures complementary structural properties related to parameter dependence, infinite divisibility, and stability within the non-commutative setting. Both have proven fundamental in the analysis of Cauchy–Stieltjes kernel (CSK) families [5,6], contributing to the characterization of variance functions (VFs), the identification of invariance and symmetry properties, and the derivation of limiting theorems associated with free and Boolean convolutions. Nevertheless, despite their significance, the t- and $T_a$-transformations have been developed and analyzed largely in isolation. A natural question that arises is whether one can develop a unified analytical framework that captures the shared structural features of these transformations while preserving their intrinsic functional symmetries. To address this, we introduce a new operator, denoted by ${\mathcal{T}}_{(a,t)}$, which brings these deformation mechanisms together within a single coherent formulation.

Let $\mathbb{P}$ represent the set of all real probability measures, and let ${\mathbb{P}}_c$ represent the subset of those having compact support. One of the central ideas in free probability is to study transformations of measures through their analytic representations, most notably via the Cauchy–Stieltjes transform (CST). For any measure $\rho \in \mathbb{P}$, this transform is defined as

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

where $\mathrm{supp}\left(\rho \right)$ represents the support of $\rho$. The function ${\mathbf{G}}_{\rho }$ plays a crucial role in non-commutative probability, as it encodes much of the analytic and algebraic structure of the measure $\rho$.

A notable transformation involving ${\mathbf{G}}_{\rho }$ was introduced by Bożejko and Wysoczański [1,2], who considered a deformation parameter $t>0$. By applying the Nevanlinna theorem, they defined a new function ${\mathbf{G}}_{{\rho }_t}$ through the relation

$${\displaystyle \frac{1}{{\mathbf{G}}_{{\rho }_t}\left(w\right)}}={\displaystyle \frac{t}{{\mathbf{G}}_{\rho }\left(w\right)}}+(1-t)w.$$

(1)

The remarkable property of this construction is that ${\mathbf{G}}_{{\rho }_t}$ is again a CST of a unique probability measure, denoted $U_t\left(\rho \right):={\rho }_t$. This operation is known as the t-deformation of $\rho$. Bożejko and Wysoczański demonstrated that $U_t\left(\rho \right)$ corresponds to the t-th additive Boolean convolution power of $\rho$, expressed as ${\rho }^{\uplus t}$ (for further information on the additive Boolean convolution ⊎, see [7]). Intuitively, this transformation interpolates between the identity map (for $t=1$) and the trivial measure (as $t\to 0$), providing a continuous deformation of probability measures within the Boolean framework.

Another transformation based on the CST, but of a different analytic form, was later investigated in [3,4,5]. For a given $\rho \in \mathbb{P}$ and a real parameter a, this transformation defines a new function via

$${\displaystyle \frac{1}{{\mathbf{G}}_{T_a\left(\rho \right)}\left(w\right)}}={\displaystyle \frac{1}{{\mathbf{G}}_{\rho }(w-a)}}+a.$$

(2)

According to the Nevanlinna theorem, the resultant function ${\mathbf{G}}_{T_a\left(\rho \right)}$ corresponds to the CST of a unique probability measure, denoted by $T_a\left(\rho \right)$ and referred to as the $T_a$-transformation of $\rho$. This map acts as a structural modification of $\rho$, preserving essential analytic properties such as positivity and normalization, while introducing a parameter shift that reflects translational or perturbation effects within free convolution semigroups.

Although the t- and $T_a$-transformations were originally developed for distinct purposes, one as a Boolean deformation and the other as a structural translation, they exhibit several common analytic and structural features. Both rely on modifications of the CST through rational combinations of ${\mathbf{G}}_{\rho }$ and the variable w, and both give rise to families of measures that are closed under their respective transformations. This observation naturally motivates the search for a unified framework that incorporates both constructions as special cases.

Expanding on this notion, we suggest a novel type of measures transformation, represented by ${\mathcal{T}}_{(a,t)}$, which combines the processes of the t- and $T_a$-deformations into a single analytic expression. For $\rho \in \mathbb{P}$, $a\in \mathbb{R}$, and $t>0$, we define

$${\displaystyle \frac{1}{{\mathbf{G}}_{{\mathcal{T}}_{(a,t)}\left(\rho \right)}\left(w\right)}}={\displaystyle \frac{t}{{\mathbf{G}}_{\rho }(w-a)}}+(1-t)w+at.$$

(3)

By virtue of the Nevanlinna theorem, the function ${\mathbf{G}}_{\mathcal{T}(a,t)}$ is again the CST of a unique probability measure, which we identify as ${\mathcal{T}}_{(a,t)}\left(\rho \right)$. This transformation unifies both classical constructions in a natural way: when $a=0$, it reduces to the Bożejko–Wysoczański t-transformation, i.e., ${\mathcal{T}}_{(0,t)}=U_t$, and when $t=1$, it coincides with the $T_a$-transformation, i.e., ${\mathcal{T}}_{(a,1)}=T_a$.

Thus, the map ${\mathcal{T}}_{(a,t)}$ provides a two-parameter generalization that smoothly interpolates between Boolean deformation and translational modification, yielding a more versatile analytic tool for studying measure transformations. This unified perspective not only simplifies the theoretical landscape of measure deformations but also opens new avenues for analyzing stability, invariance, and infinite divisibility under free and Boolean convolutions.

The effect of the ${\mathcal{T}}_{(a,t)}$-deformation on a measure $\rho \in {\mathbb{P}}_c$ can be understood particularly well through the continued-fraction representation of its CST. Any compactly supported probability measure admits a continued-fraction (Jacobi) expansion of the form

$${\displaystyle {\mathbf{G}}_{\rho }\left(w\right)=\frac{1}{{\displaystyle w-{\zeta }_1-\frac{{\iota }_1}{{\displaystyle w-{\zeta }_2-\frac{{\iota }_2}{{\displaystyle w-{\zeta }_3-\frac{{\iota }_3}{w-{\zeta }_4-\dots }}}}}}},}$$

where the coefficients ${\left({\zeta }_k\right)}_{k\ge 1}$ and ${\left({\iota }_k\right)}_{k\ge 1}$ correspond, respectively, to the recurrence (Jacobi) parameters of the orthogonal polynomials associated with $\rho$. Under ${\mathcal{T}}_{(a,t)}$-transformation, the CST of the deformed measure ${\mathcal{T}}_{(a,t)}\left(\rho \right)$ becomes

$${\displaystyle {\mathbf{G}}_{{\mathcal{T}}_{(a,t)}\left(\rho \right)}\left(w\right)=\frac{1}{{\displaystyle w-t{\zeta }_1-\frac{t{\iota }_1}{{\displaystyle w-{\zeta }_2-a-\frac{{\iota }_2}{{\displaystyle w-{\zeta }_3-a-\frac{{\iota }_3}{w-{\zeta }_4-a\dots }}}}}}}.}$$

This representation highlights how the deformation modifies the spectral structure of $\rho$ through elementary operations on its Jacobi parameters. The ${\mathcal{T}}_{(a,t)}$-operator therefore produces a controlled perturbation of the continued-fraction expansion: the parameter t scales the first “layer” of the expansion, while the parameter a shifts all subsequent layers. As a result, the transformation admits a transparent and tractable interpretation in terms of orthogonal polynomials, moments, and spectral data associated with the original measure.

A central motivation for introducing the analytical transformation ${\mathcal{T}}_{(a,t)}$ is to provide a unified framework that brings together two fundamental deformations in free probability: the t- and the $T_a$-transformations. Although each of these transformations plays an important role in the study of free convolutions, structural stability, and analytic properties of probability measures, they have traditionally been treated independently. The operator ${\mathcal{T}}_{(a,t)}$ consolidates these mechanisms into a single analytic object, allowing deformation phenomena to be analyzed in a coherent and flexible manner. This unification not only simplifies the treatment of various problems but also reveals structural features that are inaccessible through the classical transformations alone. In particular, the unified operator highlights common invariance and symmetry properties shared by key families such as the free Meixner family (FMF) and the free analog of the Letac-Mora class (FLMC), and it enables the derivation of new limiting results connecting classical free distributions. Thus, the introduction of ${\mathcal{T}}_{(a,t)}$ provides a broader and more symmetric perspective on deformation and stability phenomena in non-commutative probability theory. From a larger viewpoint, the findings of this work reveal conceptual linkages between abstract notions in free probability and well-known ideas in statistics and geometry. The CSK families and the corresponding VFs may be seen as non-commutative counterparts of Fisher information metrics and classical exponential families. This analogy offers a geometric viewpoint on deformation processes in free probability, suggesting deep connections with information geometry, quantum statistics, and complex systems modeling. Consequently, the proposed framework does more than merely generalize known transformations, it extends the theoretical frontiers of non-commutative probability and provides explicit analytic tools for studying invariance and deformation phenomena. Within this unified setting, free probability emerges as a symmetric interface among probabilistic, geometric, and statistical principles, offering new perspectives that extend from random matrix theory to the geometric foundations of statistical inference. By combining these viewpoints, the current study advances a more thorough and cohesive comprehension of the mathematical frameworks supporting contemporary theories of information and randomness.

This article presents a detailed analysis of the ${\mathcal{T}}_{(a,t)}$-deformation within the analytical setting of CSK families and their associated VFs. The investigation begins in Section 2, which provides a concise yet comprehensive review of the fundamental notions and structural properties of CSK families, ensuring that the discussion remains self-contained and accessible. In Section 3, we present a general analytical expression describing how the VF behaves under the action of the unified ${\mathcal{T}}_{(a,t)}$-transformation. This expression captures the deformation mechanism linking several important measure transformations in free probability and serves as the theoretical foundation for the results that follow. The theoretical developments are then applied, where we establish one of the central contributions of this work: the property of invariance related to the FMF and the FLMC of probability measures under the ${\mathcal{T}}_{(a,t)}$-deformation. This invariance result not only generalizes previous findings related to the t- and $T_a$-transformations but also highlights the robustness of these families within the CSK framework. Finally, Section 4 presents a series of new limiting theorems that further illustrate the analytical power of the proposed transformation. These results reveal how the ${\mathcal{T}}_{(a,t)}$-transformation can act as a unifying tool for generating new relationships among fundamental distributions in free probability. In particular, they expose a structured and elegant pathway linking several distributions such as the free binomial, the Marchenko–Pastur and the semicircle distributions, thereby deepening our understanding of the intricate landscape of free probabilistic models.

2. Basic Concepts on CSK Families

The CSK families of measures has emerged as a powerful and unifying setting within the broader landscape of noncommutative probability theory. It establishes deep conceptual parallels with the classical theory of Natural Exponential Families (NEFs), which occupy a central position in traditional statistics. In classical probability, NEFs present a systematic and elegant method for characterizing and modeling probability measures based on their moment-generating functions and the associated VFs; see [8,9,10,11]. These families have been instrumental in various fields, offering analytical tractability and a geometric interpretation that underpin major developments in statistical theory, information geometry, and Bayesian inference; see [12,13,14]. In the noncommutative setting, CSK families are constructed by substituting the exponential kernel used in NEFs with a rational kernel of the form ${(1-\vartheta l)}^{-1}$. This substitution is far from superficial, it adapts the exponential–family framework to the analytical machinery of free probability, where the CST, subordination functions, and free convolutions take the place of classical moment-based techniques. Analogous to the Laplace transform role in NEFs, the CST serves as the central analytical tool, specifically tailored to the complex-analytic structures governing free convolution and its deformations. Studying CSK families therefore provides a fresh and unifying perspective on noncommutative analogs of classical statistical models. It opens new possibilities for analyzing probabilistic and geometric structures in operator theory, random matrix theory, and free harmonic analysis. Foundational contributions by [15,16] established the analytical basis of the theory, illustrating its power to model deformation and invariance phenomena in free statistics. Subsequently, this framework was further generalized in [17] to encompass measures supported on one-sided domains, notably those with support bounded from above, thereby extending the reach and versatility of CSK families within modern free probability theory.

Let ${\mathbb{P}}_{ba}$ denote the class of non-degenerate probability measures whose support is bounded from above. For any $\rho \in {\mathbb{P}}_{ba}$, the Cauchy–Stieltjes-type moment function is defined by

$${\mathbf{M}}^{\rho }\left(\vartheta \right)=\int {\displaystyle \frac{\rho \left(dl\right)}{1-\vartheta l}}\vartheta \in [0,{\vartheta }_{+}\left(\rho \right)),$$

where $1/{\vartheta }_{+}\left(\rho \right)=\max \{\sup \mathrm{supp}\left(\rho \right),0\}$. The one-sided CSK family generated by $\rho$ is then given by

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

The mean function $\vartheta \mapsto {\mathbf{K}}^{\rho }\left(\vartheta \right)$, given by

$${\mathbf{K}}^{\rho }\left(\vartheta \right)=\int l{\mathbf{P}}_{\vartheta }^{\rho }\left(dl\right)={\displaystyle \frac{{\mathbf{M}}^{\rho }\left(\vartheta \right)-1}{\vartheta {\mathbf{M}}^{\rho }\left(\vartheta \right)}}$$

is increasing strictly on $(0,{\vartheta }_{+}\left(\rho \right))$ [17]. Its image, $(m_1\left(\rho \right),m_{+}\left(\rho \right))={\mathbf{K}}^{\rho }\left((0,{\vartheta }_{+}\left(\rho \right))\right)$, is referred to as the mean domain of the family ${\mathcal{K}}_{+}\left(\rho \right)$. Let ${\psi }^{\rho }$ denote the reciprocal of ${\mathbf{K}}^{\rho }$, one can express the mean–parameterized form of the family as

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

Moreover, if $\mathbf{B}\left(\rho \right)=1/{\vartheta }_{+}\left(\rho \right)$, then

$$m_1\left(\rho \right)=\underset{\vartheta \to 0^{+}}{\lim }{\mathbf{K}}^{\rho }\left(\vartheta \right)\mathrm{and}{m}_{+}\left(\rho \right)=\mathbf{B}\left(\rho \right)-\underset{w\to {\left(\mathbf{B}\left(\rho \right)\right)}^{+}}{\lim }1/{\mathbf{G}}_{\rho }\left(w\right).$$

If $\mathrm{supp}\left(\rho \right)$ is instead bounded from below, a symmetric construction yields the one-sided CSK family ${\mathcal{K}}_{-}\left(\rho \right)$, parameterized by $\vartheta \in ({\vartheta }_{-}\left(\rho \right),0)$, where ${\vartheta }_{-}\left(\rho \right)$ equals either $1/\mathbf{A}\left(\rho \right)$ or $-\infty$, with $\mathbf{A}\left(\rho \right)=\min \{0,\inf \mathrm{supp}(\rho \left)\right\}$. The mean domain in this case is $(m_{-}\left(\rho \right),m_1\left(\rho \right))$, where $m_{-}\left(\rho \right)=\mathbf{A}\left(\rho \right)-1/{\mathbf{G}}_{\rho }\left(\mathbf{A}\left(\rho \right)\right)$. If $\rho$ has compact support (i.e., $\rho \in {\mathbb{P}}_c$), then $\vartheta$ ranges over $({\vartheta }_{-}\left(\rho \right),{\vartheta }_{+}\left(\rho \right))$, and the complete (two-sided) CSK family is given by

$$\mathcal{K}\left(\rho \right)={\mathcal{K}}_{+}\left(\rho \right)\cup \left\{\rho \right\}\cup {\mathcal{K}}_{-}\left(\rho \right).$$

For $\rho \in {\mathbb{P}}_{ba}$, the VF associated with the family is defined as [15]

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

and plays a central role in the structure of CSK families. The VF encodes important structural information and offers a functional link between the family’s mean and variance, much like its classical equivalent in NEFs. The VF can be used to categorize CSK families, examine how stable they are under transformations, and find basic free distributions. It also acts as an analytical link between moments, generating functions, and convolution characteristics, which makes it an essential tool for free probability theory and practice.

All elements of ${\mathcal{K}}_{+}\left(\rho \right)$ have infinite variance when $\rho$ does not have a finite mean. The authors in [17] addressed this by introducing the idea of a pseudo-variance function (PVF) defined by

$${\mathbb{V}}_{\rho }\left(s\right)=s\left({\displaystyle \frac{1}{{\psi }^{\rho }\left(s\right)}}-s\right),$$

(4)

which captures second-order structure even in the absence of finite variance. When the mean $m_1\left(\rho \right)=\int l\rho \left(dl\right)$ is finite, the VF exists and is related to the PVF by

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

(5)

The PVF extends the notion of the VF to settings where the latter may not be well defined. In many CSK families, the VF requires the existence of first moment, an assumption that fails for several important distributions with heavy tails or truncated domains. The PVF bypasses this limitation by being defined analytically through raw moments. As a result, it remains meaningful even when the first moment of the generating measure do not exist. At the same time, in cases where all necessary integrability conditions are met, particularly when the generating distribution possesses a finite mean, the PVF is related to the VF. In this sense, the PVF not only generalizes the VF to a broader class of probability measures but also preserves full consistency with the classical framework whenever the usual moment assumptions hold.

Finally, several key properties summarize how these functions determine and transform the generating measure $\rho$:

• Determinacy: The PVF uniquely determines $\rho$. Specifically, if we define

  $$\varpi =\varpi \left(s\right)={\mathbb{V}}_{\rho }\left(s\right)/s+s,$$

  then

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

  (6)

  When $m_1\left(\rho \right)$ is finite, this becomes

  $${\mathbf{G}}_{\rho }\left(\varpi \right)={\displaystyle \frac{s-m_1\left(\rho \right)}{{\mathbf{V}}_{\rho }\left(s\right)}}.$$

  (7)

  Hence, ${\mathbf{V}}_{\rho }$ together with $m_1\left(\rho \right)$ fully determines $\rho$.
• Affine transformations: If $\phi \left(y\right)=\alpha y+\beta$ where $\alpha \ne 0$, then the transformed measure $\phi \left(\rho \right)$ satisfies: for s close to $m_1\left(\phi \left(\rho \right)\right)=\alpha m_1\left(\rho \right)+\beta$

  $${\mathbb{V}}_{\phi \left(\rho \right)}\left(s\right)={\displaystyle \frac{{\alpha }^{2}s}{s-\beta }}{\mathbb{V}}_{\rho }\left({\displaystyle \frac{s-\beta }{\alpha }}\right).$$

  (8)

  If $m_1\left(\rho \right)$ is finite,

  $${\mathbf{V}}_{\phi \left(\rho \right)}\left(s\right)={\alpha }^2{\mathbf{V}}_{\rho }\left({\displaystyle \frac{s-\beta }{\alpha }}\right).$$

  (9)
• When considering the $T_a$-transformation of $\rho$ defined by (2), the PVF and the VF of ${\mathcal{K}}_{+}\left(\rho \right)$ changes in the following way, see [5] (Proposition 1): For $a\in \mathbb{R}$ and for s near $m_1\left(T_a\left(\rho \right)\right)=m_1\left(\rho \right)$,

  $${\mathbb{V}}_{T_a\left(\rho \right)}\left(s\right)={\mathbb{V}}_{\rho }\left(s\right)+as.$$

  (10)

  If $m_1\left(\rho \right)$ is finite,

  $${\mathbf{V}}_{T_a\left(\rho \right)}\left(s\right)={\mathbf{V}}_{\rho }\left(s\right)+a(s-m_1\left(\rho \right)).$$

  (11)
• When considering the t-transformation of $\rho$ defined by (1), the PVF and the VF of ${\mathcal{K}}_{+}\left(\rho \right)$ changes in the following way, see [18] (Theorem 2.3): For $t>0$ and for s near $m_1\left(U_t\left(\rho \right)\right)=m_1\left({\rho }^{\uplus t}\right)=t{m}_1\left(\rho \right)$,

  $${\mathbb{V}}_{U_t\left(\rho \right)}\left(s\right)=t{\mathbb{V}}_{\rho }(s/t)+s^2(1/t-1)={\mathbb{V}}_{{\rho }^{\uplus t}}\left(s\right).$$

  (12)

  If $m_1\left(\rho \right)$ is finite,

  $${\mathbf{V}}_{U_t\left(\rho \right)}\left(s\right)=t{\mathbf{V}}_{\rho }(s/t)+s(s-t{m}_1\left(\rho \right))(1/t-1)={\mathbf{V}}_{{\rho }^{\uplus t}}\left(s\right).$$

  (13)

3. Two Families of Measures That Are Invariants Under ${\mathcal{T}}_{(\mathbf{a},\mathbf{t})}$-Transformation

Invariance under analytic transformations plays a central role in understanding the structural stability of distributional families in free probability. When a family remains unchanged under a deformation such as the ${\mathcal{T}}_{(a,t)}$-transformation, this indicates that its defining features such as its CST, VF, and underlying moment structure are preserved despite the transformation’s perturbation. Such stability properties not only reveal deep symmetries within the family but also enable a unified description of seemingly different distributions within a single analytic framework. Invariance therefore provides both a conceptual and technical foundation for identifying canonical families, understanding their robustness under deformations, and uncovering the hidden relationships that link fundamental laws in free probability.

The FMF and the FLMC occupy a central and foundational role in free probability theory due to their rich structural properties and their deep connections with classical and noncommutative probabilistic models. The FMF represents one of the most significant parametric families of freely infinitely divisible distributions, encompassing well-known examples such as the semicircle, the Marchenko–Pastur, and the free binomial laws. In random matrix theory, these distributions often appear as limit laws, especially when considering the asymptotic behavior of eigenvalue distributions of big random matrices. Their algebraic and analytic tractability makes them indispensable tools for studying free convolutions, free cumulants, and transform methods such as the R- and S-transforms. On the other hand, the FLMC extends the FMF by incorporating more general PVFs, allowing the exploration of higher-order polynomial structures within the CSK framework. This extension enables a deeper understanding of nonlinear deformation mechanisms in free probability. The FLMC captures more complex dependencies and geometric properties of noncommutative random variables, making it an essential bridge between NEFs and their free counterparts. Together, the FMF and FLMC provide a unified framework for analyzing invariance, stability, and transformation behaviors of free distributions under various operations. Their study not only advances the theoretical foundation of free probability but also has far-reaching implications for random matrix theory, free statistical inference, and information geometry, where understanding the stability and structure of these distributions leads to new insights into noncommutative analogs of classical probabilistic phenomena.

In what follows, we present a fundamental result describing how the PVF and the VF of a CSK family are modified when the generating measure undergoes the ${\mathcal{T}}_{(a,t)}$-transformation. This result forms a cornerstone for the subsequent sections, as it provides the analytic foundation upon which the main theorems of this article are established.

Proposition 1.

For $\rho \in {\mathbb{P}}_{ba}$, $t>0$ and $a\in \mathbb{R}$, consider the ${\mathcal{T}}_{(a,t)}$-deformation (3). Then, for s near $m_1\left({\mathcal{T}}_{(a,t)}\left(\rho \right)\right)=t{m}_1\left(\rho \right)$,

$${\mathbb{V}}_{{\mathcal{T}}_{(a,t)}\left(\rho \right)}\left(s\right)=t{\mathbb{V}}_{\rho }(s/t)+as+s^2(1/t-1).$$

(14)

If $m_1\left(\rho \right)$ is finite,

$${\mathbf{V}}_{{\mathcal{T}}_{(a,t)}\left(\rho \right)}\left(s\right)=t{\mathbf{V}}_{\rho }(s/t)+a(s-t{m}_1\left(\rho \right))+s(s-t{m}_1\left(\rho \right))(1/t-1).$$

(15)

Proof. 

One has

$${\mathcal{T}}_{(a,t)}\left(\rho \right)=U_t\left(T_a\left(\rho \right)\right)=T_a\left(U_t\left(\rho \right)\right).$$

(16)

Indeed:

$${\displaystyle \frac{1}{{\mathbf{G}}_{T_a\left(U_t\left(\rho \right)\right)}\left(w\right)}}={\displaystyle \frac{1}{{\mathbf{G}}_{U_t\left(\rho \right)}(w-a)}}+a={\displaystyle \frac{1}{{\mathbf{G}}_{\rho }(w-a)}}+(1-t)(w-a)+a={\displaystyle \frac{1}{{\mathbf{G}}_{{\mathcal{T}}_{(a,t)}\left(\rho \right)}\left(w\right)}},$$

and

$${\displaystyle \frac{1}{{\mathbf{G}}_{U_t\left(T_a\left(\rho \right)\right)}\left(w\right)}}={\displaystyle \frac{t}{{\mathbf{G}}_{T_a\left(\rho \right)}\left(w\right)}}+(1-t)w={\displaystyle \frac{t}{{\mathbf{G}}_{\rho }(w-a)}}+at+(1-t)w={\displaystyle \frac{1}{{\mathbf{G}}_{{\mathcal{T}}_{(a,t)}\left(\rho \right)}\left(w\right)}}.$$

Thus,

$${\mathbf{G}}_{T_a\left(U_t\left(\rho \right)\right)}\left(w\right)={\mathbf{G}}_{{\mathcal{T}}_{(a,t)}\left(\rho \right)}\left(w\right)={\mathbf{G}}_{U_t\left(T_a\left(\rho \right)\right)}\left(w\right).$$

Now, basing on (10), (12) and (16), for s near $m_1\left({\mathcal{T}}_{(a,t)}\right)=m_1\left(U_t\left(T_a\left(\rho \right)\right)\right)=t{m}_1\left(\rho \right)$, we obtain

$$\begin{array}{ccc}\hfill {\mathbb{V}}_{{\mathcal{T}}_{(a,t)}\left(\rho \right)}\left(s\right)={\mathbb{V}}_{U_t\left(T_a\left(\rho \right)\right)}\left(s\right)& =& t{\mathbb{V}}_{T_a\left(\rho \right)}(s/t)+s^2(1/t-1)\hfill \\ & =& t\left[{\mathbb{V}}_{\rho }(s/t)+as/t\right]+s^2(1/t-1)\hfill \\ & =& t{\mathbb{V}}_{\rho }(s/t)+as+s^2(1/t-1).\hfill \end{array}$$

which is nothing but relation (14). If $m_1\left(\rho \right)$ is finite, relation (15) is obtained by combining (5) and (14). □

Proposition 1 provides the precise transformation rules for the PVF and VF of any CSK family, and these rules are exactly what allow us to identify when a family remains unchanged (i.e., invariant) under ${\mathcal{T}}_{(a,t)}$. The subsequent invariance theorems for the FMF (Theorem 1) and for the FLMC (Theorem 2) rely directly on these transformation formulas. In other words, Proposition 1 is not only a technical result but also the key tool that reveals the structural conditions under which invariance occurs.

3.1. Invariance of the FMF Under ${\mathcal{T}}_{(a,t)}$-Transformation

The quadratic class of CSK families is one of the most fundamental and well-studied cases within the theory of CSK families. It is characterized by a quadratic VF of the form

$${\mathbf{V}}_{\rho }\left(s\right)=1+a_{1}s+a_2{s}^2,a_1\in \mathbb{R},a_2\ge -1,m_1\left(\rho \right)=0$$

(17)

as established in [15]. This elegant form of the VF gives rise to a broad family of probability measures known as the FMF. The general expression for the measures in this family is given by

$$\rho \left(dl\right)={\displaystyle \frac{\sqrt{4(1+a_2)-{(l-a_1)}^2}}{2\pi (a_2{l}^2+a_{1}l+1)}}{1}_{(a_1-2\sqrt{1+a_2},a_1+2\sqrt{1+a_2})}\left(l\right)dl+p_1{\delta }_{l_1}+p_2{\delta }_{l_2}.$$

(18)

where the continuous part is supported on a compact interval, and depending on the parameters $(a_1,a_2)$, additional atomic parts may appear. Different parameter regimes lead to specific cases of the FMF:

(i)

If $a_2=0,a_1^2>1$, then $p_1=1-1/a_1^2,l_1=-1/a_1,p_2=0$.

(ii)

If $a_2>0$ and $a_1^2>4a_2$, then $p_1=\max \{0,1-{\displaystyle \frac{|a_1|-\sqrt{a_1^2-4a_2}}{2a_2\sqrt{a_1^2-4a_2}}}\},p_2=0$, and its location is given by $l_1=\pm {\displaystyle \frac{|a_1|-\sqrt{a_1^2-4a_2}}{2a_2}}$ with the sign opposite to the sign of $a_1$.

(iii)

If $-1\le a_2<0$, two atoms may appear at

$$l_{1,2}={\displaystyle \frac{-a_1\pm \sqrt{a_1^2-4a_2}}{2a_2}},p_{1,2}=1+{\displaystyle \frac{\sqrt{a_1^2-4a_2}\mp a_1}{2a_2\sqrt{a_1^2-4a_2}}}$$

These parameter-dependent cases encompass several well-known free probability distributions, demonstrating how the FMF unifies them into a single analytical framework. In particular, up to scaling (dilation) and free convolution, the distribution $\rho$ corresponds to:

(i)

The semicircle ($\mathcal{SC}$) distribution if $a_1=a_2=0$.

(ii)

The Marchenko–Pastur ($\mathcal{MP}$) distribution if $a_2=0$ and $a_1\ne 0$.

(iii)

The free Pascal ($\mathcal{FP}$) distribution if $a_2>0$ and $a_1^2>4a_2$.

(iv)

The free Gamma ($\mathcal{FG}$) distribution if $a_2>0$ and $a_1^2=4a_2$.

(v)

The free analog of hyperbolic ($\mathcal{FH}$) distribution if $a_2>0$ and $a_1^2<4a_2$.

(vi)

The free binomial ($\mathcal{FB}$) distribution if $-1\le a_2<0$.

This classification highlights the rich diversity encapsulated within the quadratic CSK families. In what follows, we demonstrate a key structural property of this family: the FMF (which correspond to quadratic CSK families) remains invariant under the ${\mathcal{T}}_{(a,t)}$-transformation. This means that applying this transformation to a measure from the FMF yields another measure within the same family, up to possible dilation, denoted by ${\mathbf{D}}_{\alpha }\left(\rho \right)$ where ${\mathbf{D}}_{\alpha }$ represents scaling of the measure $\rho$ by nonzero factor $\alpha$.

Theorem 1.

If $\rho \in \mathrm{FMF}$, then ∀ $t>0$ and ∀ $a\in \mathbb{R}$, ${\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(\rho \right)\right)\in \mathrm{FMF}$.

Proof. 

If $\rho \in \mathrm{FMF}$, then the VF of $\mathcal{K}\left(\rho \right)$ is

$${\mathbf{V}}_{\rho }\left(s\right)=1+a_1^{\prime }s+a_2^{\prime }{s}^2,a_1^{\prime }\in \mathbb{R},a_2^{\prime }\ge -1.$$

(19)

Relations (9), (15) and (19) gives

$${\mathbf{V}}_{{\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(\rho \right)\right)}\left(s\right)=1+s\left(a_1^{\prime }\sqrt{t}+a\right)+s^2\left(t(a_2^{\prime }+1)-1\right),s\mathrm{close}\mathrm{to}{m}_1\left({\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(\rho \right)\right)\right)=0$$

(20)

which corresponds to a VF of the from given in (17). Then, ${\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(\rho \right)\right)\in \mathrm{FMF}$. □

Theorem 1 demonstrates a remarkable structural property: the FMF remains closed under the action of the ${\mathcal{T}}_{(a,t)}$-operator. This finding reveals a deep form of stability and internal consistency within the FMF, showing that its analytical form and probabilistic characteristics are preserved even under non-trivial deformations. Such robustness emphasizes the central position of the FMF in the broader landscape of non-commutative probability, where invariance properties often signify deep algebraic and geometric symmetries. Moreover, this result provides a strong theoretical foundation for extending classification schemes and exploring hierarchies among free distributions. It suggests that the FMF acts as a closed and self-contained structure under fundamental transformations, making it a natural candidate for modeling invariant phenomena in free probability and related fields such as random matrix theory.

To further highlight the significance and applicability of this Theorem, the following examples concretely demonstrate how the ${\mathcal{T}}_{(a,t)}$-transformation operates within the FMF and how it preserves key statistical and analytic properties of the associated measures.

Corollary 1.

Consider $sb\left(dl\right)={\displaystyle \frac{1}{2}}{\delta }_{-1}+{\displaystyle \frac{1}{2}}{\delta }_1$. Then, ${\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(sb\right)\right)$ is a $\mathcal{FB}$ distribution (18) with $a_1=a$ and $a_2=-1$.

Proof. 

The VF of $\mathcal{K}\left(sb\right)$ is of the kind (19) for $a_1^{\prime }=0$ and $a_2^{\prime }=-1$. From (20) one has

$${\mathbf{V}}_{{\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(sb\right)\right)}\left(s\right)=1+sa-s^2,s\mathrm{near}0.$$

In comparison with (17), this result completes the proof through the application of (7). □

Corollary 2.

Consider the $\mathcal{SC}$ distribution

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

(21)

Then, ${\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(sc\right)\right)$ is a

(i) 

$\mathcal{SC}$ distribution, if $a=0$ and $t=1$.

(ii) 

$\mathcal{MP}$ distribution with $a_1=a$ and $a_2=0$ if, $a\ne 0$ and $t=1$.

(iii) 

$\mathcal{FP}$ distribution with $a_1=a$ and $a_2=t-1$, if $t>1$ and $a^2>4(t-1)$.

(iv) 

$\mathcal{FG}$ distribution with $a_1=a$ and $a_2=t-1$, if $t>1$ and $a^2=4(t-1)$.

(v) 

$\mathcal{FH}$ distribution with $a_1=a$ and $a_2=t-1$, if $t>1$ and $a^2<4(t-1)$,

(vi) 

$\mathcal{FB}$ distribution with $a_1=a$ and $a_2=t-1$, if $0<t<1$.

Proof. 

The VF of $\mathcal{K}\left(sc\right)$ is of the form (19) with $a_1^{\prime }=a_2^{\prime }=0$. Relation (20) gives

$${\mathbf{V}}_{{\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(sc\right)\right)}\left(s\right)=1+sa+s^2\left(t-1\right),s\mathrm{near}0.$$

(22)

By comparing (22) with (17), we obtain the following results:

(i)

If $a=0$ and $t=1$, then ${\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(sc\right)\right)=sc$.

(ii)

If $a\ne 0$ and $t=1$, then ${\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(sc\right)\right)$ is a $\mathcal{MP}$ distribution (18) with $a_1=a$ and $a_2=0$.

(iii)

If $t>1$ and $4(t-1)<a^2$, then ${\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(sc\right)\right)$ is a $\mathcal{FP}$ distribution (18) with $a_1=a$ and $a_2=t-1$.

(iv)

If $t>1$ and $4(t-1)=a^2$, then ${\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(sc\right)\right)$ is a $\mathcal{FG}$ distribution (18) with $a_1=a$ and $a_2=t-1$.

(v)

If $t>1$ and $4(t-1)>a^2$, then ${\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(sc\right)\right)$ is a $\mathcal{FH}$ distribution (18) with $a_1=a$ and $a_2=t-1$.

(vi)

If $t\in (0,1)$, then ${\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(sc\right)\right)$ is a $\mathcal{FB}$ distribution (18) with $a_1=a$ and $a_2=t-1$.

 □

Corollary 3.

For $a_1^{\prime }\ne 0$ and $a_2^{\prime }=0$, consider the $\mathcal{MP}$ distribution

$$m{p}_{a_1^{\prime }}\left(dl\right)={\displaystyle \frac{\sqrt{4-{(l-a_1^{\prime })}^2}}{2\pi (a_1^{\prime }l+1)}}{1}_{(a_1^{\prime }-2,a_1^{\prime }+2)}\left(l\right)dl+{(1-1/a_1^{\prime 2})}^{+}{\delta }_{-1/a_1^{\prime }}.$$

(23)

Then, ${\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(m{p}_{a_1^{\prime }}\right)\right)$ is a

(i) 

$\mathcal{SC}$ distribution with $a_1=a_1^{\prime }+a=0$ and $a_2=a_2^{\prime }=0$ if $t=1$ and $a=-a_1^{\prime }$.

(ii) 

$\mathcal{MP}$ distribution with $a_1=a_1^{\prime }+a$ and $a_2=a_2^{\prime }=0$ if $t=1$ and $a\ne -a_1^{\prime }$.

(iii) 

$\mathcal{FP}$ distribution with $a_1=a_1^{\prime }\sqrt{t}+a$ and $a_2=t-1$, if $t>1$ and ${(a_1^{\prime }\sqrt{t}+a)}^2>4(t-1)$.

(iv) 

$\mathcal{FG}$ distribution with $a_1=a_1^{\prime }\sqrt{t}+a$ and $a_2=t-1$, if $t>1$ and ${(a_1^{\prime }\sqrt{t}+a)}^2=4(t-1)$.

(v) 

$\mathcal{FH}$ distribution with $a_1=a_1^{\prime }\sqrt{t}+a$ and $a_2=t-1$, if $t>1$ and ${(a_1^{\prime }\sqrt{t}+a)}^2<4(t-1)$.

(vi) 

$\mathcal{FB}$ distribution with $a_1=a_1^{\prime }\sqrt{t}+a$ and $a_2=t-1$, if $0<t<1$.

Proof. 

The VF of $\mathcal{K}\left(m{p}_{a_1^{\prime }}\right)$ is of the kind (19) with $a_1^{\prime }\ne 0$ and $a_2^{\prime }=0$. Using (20), we get

$${\mathbf{V}}_{{\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(m{p}_{a_1^{\prime }}\right)\right)}\left(s\right)=1+s(a_1^{\prime }\sqrt{t}+a)+s^2\left(t-1\right),s\mathrm{close}\mathrm{to}0.$$

(24)

Relations (17) and (24) gives:

(i)

If $a=-a_1^{\prime }$ and $t=1$, then ${\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(m{p}_{a_1^{\prime }}\right)\right)$ is a $\mathcal{SC}$ distribution (18) with $a_1=a_1^{\prime }+a=0$ and $a_2=0$.

(ii)

If $a\ne -a_1^{\prime }$ and $t=1$, then ${\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(m{p}_{a_1^{\prime }}\right)\right)$ is a $\mathcal{MP}$ distribution (18) with $a_1=a_1^{\prime }+a$ and $a_2=0$.

(iii)

If $t>1$ and ${(a_1^{\prime }\sqrt{t}+a)}^2>4(t-1)$, then ${\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(m{p}_{a_1^{\prime }}\right)\right)$ is a $\mathcal{FP}$ distribution (18) with $a_1=a_1^{\prime }\sqrt{t}+a$ and $a_2=t-1$.

(iv)

If $t>1$ and ${(a_1^{\prime }\sqrt{t}+a)}^2=4(t-1)$, then ${\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(m{p}_{a_1^{\prime }}\right)\right)$ is a $\mathcal{FG}$ distribution (18) with $a_1=a_1^{\prime }\sqrt{t}+a$ and $a_2=t-1$.

(v)

If $t>1$ and ${(a_1^{\prime }\sqrt{t}+a)}^2<4(t-1)$, then ${\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(m{p}_{a_1^{\prime }}\right)\right)$ is a $\mathcal{FH}$ distribution (18) with $a_1=a_1^{\prime }\sqrt{t}+a$ and $a_2=t-1$.

(vi)

If $t\in (0,1)$, then ${\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(m{p}_{a_1^{\prime }}\right)\right)$ is a $\mathcal{FB}$ distribution (18) with $a_1=a_1^{\prime }\sqrt{t}+a$ and $a_2=t-1$.

 □

A remarkable property of the ${\mathcal{T}}_{(a,t)}$-transformation is its ability, under suitable parameter choices, to map the $\mathcal{MP}$ distribution onto the $\mathcal{SC}$ distribution. This transformation establishes a direct analytical bridge between two of the most fundamental laws in free probability theory. What makes this feature particularly striking is that such a correspondence cannot be achieved using the t-transformation alone. The t-transformation deforms measures along a single direction of Boolean-type convolution, whereas the ${\mathcal{T}}_{(a,t)}$-transformation introduces an additional degree of freedom through the parameter a, allowing for a richer and more flexible class of deformations. This new link between the $\mathcal{MP}$ and $\mathcal{SC}$ laws highlights the unifying strength and analytical depth of the ${\mathcal{T}}_{(a,t)}$-framework. It shows that the transformation is not merely a generalization of existing operations but a tool capable of revealing hidden structural connections between seemingly distinct free distributions. In doing so, it opens new avenues for exploring the interplay between free additive and multiplicative convolution models and for developing deeper probabilistic and geometric interpretations within the CSK family framework.

3.2. Invariance of the FLMC Under ${\mathcal{T}}_{(a,t)}$-Transformation

In [17], the authors provided an explicit characterization of a class of cubic CSK families that can be described through their PVF. These families are defined by the relation

$${\mathbb{V}}_{\rho }\left(s\right)=s(b_1{s}^2+b_{2}s+b_3),m_1\left(\rho \right)=-\infty$$

(25)

where the parameter $b_1$ is strictly positive, ensuring the cubic growth behavior of the PVF. This expression generalizes the structure of quadratic CSK families and represents a natural extension that captures more complex dependencies between the mean and the variance–like function.

The probability measures corresponding to these cubic families take the following analytical form:

$$\rho \left(dl\right)={\displaystyle \frac{\sqrt{4b_1{b}_3-{(b_2+1)}^2-4b_{1}l}}{2\pi (b_3+b_{2}l+b_1{l}^2)}}{\mathbf{1}}_{\left(-\infty ,\rho -{\displaystyle \frac{{(b_2+1)}^2}{\left(4b_1\right)}}\right)}\left(l\right)dl+p(b_1,b_2,b_3){\delta }_{-(b_2+\sqrt{b_2^2-4b_1{b}_3})/\left(2b_1\right)},$$

(26)

where the term involving the Dirac mass accounts for the presence of an atom in the distribution, depending on the parameter configuration. The weight of this atomic component is given by

$$p(b_1,b_2,b_3)=\left\{\begin{array}{cc}1-1/\sqrt{b_2^2-4b_1{b}_3},\hfill & if{b}_2^2>4b_1{b}_3+1;\hfill \\ 0,\hfill & otherwise.\hfill \end{array}\right.$$

This construction reveals that cubic CSK families encompass both absolutely continuous and discrete parts, depending on the interplay between $b_1$, $b_2$, and $b_3$. The continuous component describes a density supported on a semi-infinite interval determined by the parameters, while the discrete mass emerges only when the discriminant condition $b_2^2>4b_1{b}_3+1$ is satisfied. These cubic families thus provide a rich and flexible framework that extends beyond the quadratic case, allowing for the modeling of more intricate behaviors within the theory of free probability and noncommutative statistics.

Among the cubic CSK families, one of the most remarkable and instructive examples is the inverse semicircle (ISC) law, which is defined by the density

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

(27)

This distribution stands out due to its distinctive analytic form and its deep connection to the semicircle law, one of the cornerstone measures in free probability. The ISC law plays a fundamental role in illustrating the behavior of cubic CSK families and serves as a canonical representative of this class.

For this distribution, the associated PVF takes the particularly simple and elegant form

$${\mathbb{V}}_{isc}\left(s\right)=s^3,s\in (m_1\left(isc\right),m_{+}\left(isc\right))=(-\infty ,-1).$$

(28)

In terms of parameters, the ISC law corresponds to the specific setting $b_1=1$, $b_2=0$, and $b_3=0$ in Equation (26). This configuration marks the simplest yet most illuminating example within the cubic family, as it captures the essence of the cubic structure while maintaining analytical tractability.

Furthermore, the comprehensive description and classification of cubic CSK families, including the ISC law and other notable distributions, are elaborated in [17], particularly in Section 4. That section provides a detailed account of the FLMC of probability measures described in Equation (26), offering deeper insights into the connection between classical NEFs and their free counterparts within the CSK framework.

In what follows, we present a key result that characterizes the behavior of measures within FLMC and establishes fundamental connections between this class and the ${\mathcal{T}}_{(a,t)}$-transformation. This result will serve as the foundation for understanding the invariance and transformation properties of cubic CSK families.

Theorem 2.

If $\rho \in$ FLMC, then for every $t>0$ and every $a\in \mathbb{R}$, ${\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(\rho \right)\right)\in$ FLMC.

Proof. 

If $\rho \in$ FLMC, then the PVF of ${\mathcal{K}}_{+}\left(\rho \right)$ may be written as

$${\mathbb{V}}_{\rho }\left(s\right)=s(b_1^{\prime }{s}^2+b_2^{\prime }s+b_3^{\prime }),b_1^{\prime }>0,b_2^{\prime },b_3^{\prime }\in \mathbb{R}.$$

(29)

Equations (8), (14) and (29) gives,

$${\mathbb{V}}_{{\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(\rho \right)\right)}\left(s\right)=s(b_1^{\prime }t\sqrt{t}{s}^2+(b_2^{\prime }t+t-1)s+b_3^{\prime }\sqrt{t}+a),s\mathrm{close}\mathrm{to}{m}_1\left({\mathcal{T}}_{(a,1/t)}\left({\mathbf{D}}_{\sqrt{t}}\left(\rho \right)\right)\right)=-\infty$$

(30)

which correspond to a cubic PVF of the kind specified in (25). So, ${\mathcal{T}}_{(a,1/t)}\left(D_{\sqrt{t}}\left(\rho \right)\right)$ belongs to FLMC. □

The cubic invariance result shows that when the PVF of a CSK family is a polynomial of degree three, its polynomial structure is preserved under the ${\mathcal{T}}_{(a,t)}$-transformation. In other words, the transformation does not increase or reduce the degree of the underlying polynomial but merely modifies its coefficients in a controlled analytic manner. This preservation of polynomial degree is significant because it highlights a structural rigidity of the family: cubic CSK families remain within the same class after deformation, indicating that they form a stable and naturally closed category under ${\mathcal{T}}_{(a,t)}$. Such behavior reflects deep algebraic symmetry and reinforces the idea that the transformation acts as a genuine structural deformation rather than altering the fundamental complexity of the family.

Corollary 4.

Consider the ISC law (27) with $m_1\left(isc\right)=-\infty$. Then, ${\mathcal{T}}_{(1,1)}\left(isc\right)$ is the free strict arcsine distribution (26) with $b_1=1$, $b_2=0$ and $b_3=1$.

Proof. 

Relations (28) and (30) gives

$${\mathbb{V}}_{{\mathcal{T}}_{(1,1)}\left(isc\right)}\left(s\right)=s(s^2+1),s\mathrm{close}\mathrm{to}{m}_1\left({\mathcal{T}}_{(1,1)}\left(isc\right)\right)=-\infty$$

which corresponds to the PVF of the CSK family generated by the free strict arcsine distribution, see [17] (p. 590). This completes the proof through the application of (6). □

4. Limit Laws Arising from the ${\mathcal{T}}_{(\mathbf{a},\mathbf{t})}$-Operator

In this section, we introduce new limiting theorems that involve the ${\mathcal{T}}_{(a,t)}$-operator in connection with two key operations in free probability theory: the additive free convolution ⊞ (see [19,20]) and the Belinschi-Nica map [21].

The Belinschi-Nica map, denoted by ${\mathbb{B}}_p$, acts on the space of probability measures $\mathbb{P}$ according to

$$\begin{array}{ccc}\hfill {\mathbb{B}}_p:\mathbb{P}& \to & \mathbb{P};\rho \mapsto {\mathbb{B}}_p\left(\rho \right)={\left({\rho }^{⊞(p+l)}\right)}^{\uplus {\displaystyle \frac{1}{p+l}}},p\ge 0.\hfill \end{array}$$

This operation generalizes several well-known mappings between free and Boolean convolutions. In particular, for $p=1$, ${\mathbb{B}}_1$ coincides with the celebrated Bercovici-Pata bijection, which provides a correspondence between freely and classically infinitely divisible distributions.

Let $\rho \in {\mathbb{P}}_{ba}$ be a measure with a finite first moment. For any $\lambda >0$ such that the free additive convolution power ${\rho }^{⊞\lambda }$ is well-defined, the associated VF satisfies the relation [17]:

$${\mathbf{V}}_{{\rho }^{⊞\lambda }}\left(s\right)=s{\mathbf{V}}_{\rho }(s/\lambda )$$

(31)

for all s sufficiently close to the mean $m_1\left(\rho \right)$. This scaling identity reveals how the VF behaves under convolution powers, providing an important analytical tool for studying limit transitions. Moreover, as shown in [18], for $p\ge 0$ and for s near $m_1\left(\rho \right)$,

$${\mathbf{V}}_{{\mathbb{B}}_p\left(\rho \right)}\left(s\right)={\mathbf{V}}_{\rho }\left(s\right)+ps(s-m_1\left(\rho \right))\mathrm{and}{\mathbf{V}}_{{\mathbb{B}}_1^{-1}\left(\rho \right)}\left(s\right)={\mathbf{V}}_{\rho }\left(s\right)-s(s-m_1\left(\rho \right)).$$

(32)

These relations demonstrate the additive deformation effect induced by the Belinschi-Nica map on the VF, highlighting its deep algebraic connection with CSK families.

Building upon these foundations, we now proceed to state and demonstrate the main results of this section. These findings reveal new asymptotic behaviors under the joint action of the ${\mathcal{T}}_{(a,t)}$-operator, the Belinschi-Nica transformation and the free additive convolution, thus deepening the interaction between analytic transform techniques and probabilistic structures within the framework of free probability theory.

Theorem 3.

Let $\rho \in {\mathbb{P}}_c$. Then

(i) 

For any real number a and any strictly positive real number t, so that ${\mathcal{T}}_{(a,t)}\left({\rho }^{⊞1/t}\right)$ is defined, we have:

$${\mathcal{T}}_{(a,t)}\left({\rho }^{⊞1/t}\right)\stackrel{t\to +\infty }{\to }{T}_a\left({\mathbb{B}}_1^{-1}\left(\rho \right)\right),in distribution.$$

(33)

(ii) 

For any real number a and any strictly positive real number t, so that ${\left({\mathcal{T}}_{(a,1/t)}\left(\rho \right)\right)}^{⊞t}$ is defined, we have:

$${\left({\mathcal{T}}_{(a,1/t)}\left(\rho \right)\right)}^{⊞t}\stackrel{t\to +\infty }{\to }{\mathbb{B}}_1\left(T_a\left(\rho \right)\right),in distribution.$$

(34)

Proof. 

(i) For s near $m_1\left({\mathcal{T}}_{(a,t)}\left({\rho }^{⊞1/t}\right)\right)=m_1\left(\rho \right)$, relations (31) and (15) gives

$$\begin{array}{ccc}\hfill {\mathbf{V}}_{{\mathcal{T}}_{(a,t)}\left({\rho }^{⊞1/t}\right)}\left(s\right)& =& t{\mathbf{V}}_{{\rho }^{⊞1/t}}\left(s/t\right)+a\left(s-t{m}_1\left({\rho }^{⊞1/t}\right)\right)+s\left(1/t-1\right)\left(s-t{m}_1\left({\rho }^{⊞1/t}\right)\right)\hfill \\ & =& {\mathbf{V}}_{\rho }\left(s\right)+a(s-m_1\left(\rho \right))+s(s-m_1\left(\rho \right))(1/t-1)\hfill \\ & \stackrel{t\to +\infty }{\to }& {\mathbf{V}}_{\rho }\left(s\right)+a(s-m_1\left(\rho \right))-s\left(s-m_1\left(\rho \right)\right).\hfill \end{array}$$

(35)

On the other hand, for s near $m_1\left(T_a\left({\mathbb{B}}_1^{-1}\left(\rho \right)\right)\right)=m_1\left(\rho \right)$, relations (32) and (11) gives

$${\mathbf{V}}_{T_a\left({\mathbb{B}}_1^{-1}\left(\rho \right)\right)}\left(s\right)={\mathbf{V}}_{{\mathbb{B}}_1^{-1}\left(\rho \right)}\left(s\right)+a(s-m_1\left(\rho \right))={\mathbf{V}}_{\rho }\left(s\right)+a(s-m_1\left(\rho \right))-s\left(s-m_1\left(\rho \right)\right).$$

(36)

Relations (35) and (36) gives

$${\mathbf{V}}_{{\mathcal{T}}_{(a,t)}\left({\rho }^{⊞1/t}\right)}\left(s\right)\stackrel{t\to +\infty }{\to }{\mathbf{V}}_{T_a\left({\mathbb{B}}_1^{-1}\left(\rho \right)\right)}\left(s\right),s\mathrm{near}{m}_1\left(\rho \right).$$

This fact, when considered together with Proposition 4.2 from [15], completes the proof of Equation (33).

(ii) For s near $m_1\left({\left({\mathcal{T}}_{(a,1/t)}\left(\rho \right)\right)}^{⊞t}\right)=m_1\left(\rho \right)$, relations (31) and (15) gives

$$\begin{array}{ccc}\hfill {\mathbf{V}}_{{\left({\mathcal{T}}_{(a,1/t)}\left(\rho \right)\right)}^{⊞t}}\left(s\right)& =& t{\mathbf{V}}_{{\mathcal{T}}_{(a,1/t)}\left(\rho \right)}\left({\displaystyle \frac{s}{t}}\right)\hfill \\ & =& t\left[{\displaystyle \frac{1}{t}}{\mathbf{V}}_{\rho }\left(s\right)+a\left({\displaystyle \frac{s}{t}}-{\displaystyle \frac{m_1\left(\rho \right)}{t}}\right)+{\displaystyle \frac{s}{t}}\left({\displaystyle \frac{s}{t}}-{\displaystyle \frac{m_1\left(\rho \right)}{t}}\right)(t-1)\right]\hfill \\ & =& {\mathbf{V}}_{\rho }\left(s\right)+a(s-m_1\left(\rho \right))+s(s-m_1\left(\rho \right))(1-1/t)\hfill \\ & \stackrel{t\to +\infty }{\to }& {\mathbf{V}}_{\rho }\left(s\right)+a(s-m_1\left(\rho \right))+s(s-m_1\left(\rho \right)).\hfill \end{array}$$

(37)

On the other hand, for s near $m_1\left({\mathbb{B}}_1\left(T_a\left(\rho \right)\right)\right)=m_1\left(\rho \right)$, relations (32) and (11) gives,

$${\mathbf{V}}_{{\mathbb{B}}_1\left(T_a\left(\rho \right)\right)}\left(s\right)={\mathbf{V}}_{T_a\left(\rho \right)}\left(s\right)+s(s-m_1\left(T_a\left(\rho \right)\right))={\mathbf{V}}_{\rho }\left(s\right)+a(s-m_1\left(\rho \right))+s(s-m_1\left(\rho \right)).$$

(38)

Equations (37) and (38) gives

$${\mathbf{V}}_{{\left({\mathcal{T}}_{(a,1/t)}\left(\rho \right)\right)}^{⊞t}}\left(s\right)\stackrel{t\to +\infty }{\to }{\mathbf{V}}_{{\mathbb{B}}_1\left(T_a\left(\rho \right)\right)}\left(s\right),s\mathrm{near}{m}_1\left(\rho \right).$$

This fact, when considered together with Proposition 4.2 from [15], completes the proof of Equation (34). □

Next, we emphasize the significance of Theorem 3 by illustrating how it can be effectively applied to several specific and well-known probability measures. Through these examples, we aim to demonstrate the practical reach and interpretative power of the Theorem within the framework of free probability. Each application provides deeper insight into how the Theorem captures structural relationships between measures and explains their transformations under the ${\mathcal{T}}_{(a,t)}$-operation. In doing so, we not only validate the theoretical result but also reveal its potential to unify and extend existing findings concerning fundamental distributions in the field.

Example 1.

For any real number a, we have

$${\mathcal{T}}_{(a,t)}\left({sc}^{⊞1/t}\right)\stackrel{t\to +\infty }{\to }{fb}_{(a,-1)},in distribution,$$

where $sc$ is the $\mathcal{SC}$ distribution (21) and ${fb}_{(a_1,a_2)}$ represents the $\mathcal{FB}$ distribution (18) with $a_1\in \mathbb{R}$ and $-1\le a_2<0$. This is due to the fact that: for s near 0,

$$\begin{array}{ccc}\hfill {\mathbf{V}}_{{\mathcal{T}}_{(a,t)}\left({sc}^{⊞1/t}\right)}\left(s\right)\stackrel{t\to +\infty }{\to }{\mathbf{V}}_{T_a\left({\mathbb{B}}_1^{-1}\left(sc\right)\right)}\left(s\right)& =& {\mathbf{V}}_{{\mathbb{B}}_1^{-1}\left(sc\right)}\left(s\right)+as\hfill \\ & =& {\mathbf{V}}_{sc}\left(s\right)-s^2+as=1+as-s^2={\mathbf{V}}_{{fb}_{(a,-1)}}\left(s\right).\hfill \end{array}$$

Example 2.

For $a\in \mathbb{R}\setminus \left\{0\right\}$, one see that

$${\left({\mathcal{T}}_{(a,1/t)}\left(sb\right)\right)}^{⊞t}\stackrel{t\to +\infty }{\to }{mp}_a,in distribution,$$

where $sb$ is the Symmetric Bernoulli distribution and ${mp}_a$ represents the $\mathcal{MP}$ distribution (23) with $a_1^{\prime }=a$. This is due to the fact that: for s close to 0,

$$\begin{array}{ccc}\hfill {\mathbf{V}}_{{\left({\mathcal{T}}_{(a,1/t)}\left(sb\right)\right)}^{⊞t}}\left(s\right)\stackrel{t\to +\infty }{\to }{\mathbf{V}}_{{\mathbb{B}}_1\left(T_a\left(sb\right)\right)}\left(s\right)& =& {\mathbf{V}}_{T_a\left(sb\right)}\left(s\right)+s^2\hfill \\ & =& {\mathbf{V}}_{sb}\left(s\right)+as+s^2\hfill \\ & =& 1-s^2+as+s^2=1+as={\mathbf{V}}_{{mp}_a}\left(s\right).\hfill \end{array}$$

Example 3.

For any real number a, one has

$${\mathcal{T}}_{(a,t)}\left({mp}_{a_1^{\prime }}^{⊞1/t}\right)\stackrel{t\to +\infty }{\to }{fb}_{(a_1^{\prime }+a,-1)},in distribution.$$

This is due to the fact that: for s close to 0,

$$\begin{array}{ccc}\hfill {\mathbf{V}}_{{\mathcal{T}}_{(a,t)}\left({mp}_{a_1^{\prime }}^{⊞1/t}\right)}\left(s\right)\stackrel{t\to +\infty }{\to }{\mathcal{V}}_{T_a\left({\mathbb{B}}_1^{-1}\left({mp}_{a_1^{\prime }}\right)\right)}\left(s\right)& =& {\mathbf{V}}_{{\mathbb{B}}_1^{-1}\left({mp}_{a_1^{\prime }}\right)}\left(s\right)+as\hfill \\ & =& {\mathbf{V}}_{{mp}_{a_1^{\prime }}}\left(s\right)-s^2+as\hfill \\ & =& 1+(a_1^{\prime }+a)s-s^2={\mathbf{V}}_{{fb}_{(a_1^{\prime }+a,-1)}}\left(s\right).\hfill \end{array}$$

The limiting theorems employ the ${\mathcal{T}}_{(a,t)}$-deformation as a powerful analytical tool to establish explicit links between several cornerstone distributions of free probability theory, namely the $\mathcal{SC}$, the $\mathcal{MP}$, and the $\mathcal{FB}$ laws. Through this approach, we uncover how these fundamental measures are interconnected within a broader and more coherent probabilistic framework. The ${\mathcal{T}}_{(a,t)}$-transformation thus serves not merely as a mapping but as a mechanism that systematically explains transitions and deformations between these distributions. This insight reveals an underlying hierarchical organization among free probabilistic models, one that could not be observed by analyzing the $T_a$-transformation in isolation. Consequently, the results shed new light on the structural relationships and asymptotic behaviors that unify diverse families of measures within noncommutative probability.

Merkle [22] provides a comprehensive overview of the weak convergence of probability measures, detailing both theoretical foundations and practical criteria for convergence in distribution. This work serves as a cornerstone in probability theory, particularly in understanding the limiting behavior of sequences of random variables. In the context of free probability and CSK families, the concept of weak convergence is directly relevant when studying the asymptotic limits of deformed distributions under transformations such as ${\mathcal{T}}_{(a,t)}$, while classical applications of weak convergence typically focus on real–valued random variables and their empirical distributions, our work extends these ideas to non-commutative probability spaces, examining how invariance properties and limiting distributions emerge under analytic deformations. This comparison highlights that, although the underlying mathematical framework differs, the same convergence principles govern the stability and asymptotic behavior of distributions, bridging classical probability theory and its free–probabilistic counterparts.

5. Conclusions

This study demonstrates the extraordinary potential of CSK families and VFs as a strong analytical setting for investigating probability measure deformations. We invented and carefully explored the ${\mathcal{T}}_{(a,t)}$-operator, which unifies the t- and the $T_a$-deformations. By describing this transformation using the CST, we created a rigorous analytic framework that neatly generalizes these two previously separate notions. Our examination from the lens of CSK families has been very instructive. We found a generic equation for the VF of a CSK family induced by a distribution changed using ${\mathcal{T}}_{(a,t)}$-operator. This discovery provides a comprehensive computational tool for investigating and quantifying the influence of the transformation in real analytical scenarios. Using this formula, we proved a major theoretical finding: the FMF and FLMC remain invariant under the ${\mathcal{T}}_{(a,t)}$-deformation. This invariance quality highlights the structural coherence of these families within the algebraic and analytic processes of free probability.

Beyond these structural findings, our study also yielded new limiting theorems that demonstrate the ability of the ${\mathcal{T}}_{(a,t)}$-transformation to uncover deep interconnections between major distributions in free probability. In particular, we showed how the transformation constructs explicit bridges between the $\mathcal{FB}$, $\mathcal{MP}$, and $\mathcal{SC}$ laws. These relationships expose a unified and hierarchical framework among the cornerstone measures of free probability, a structure that had remained hidden when studying the $T_a$- or t-transformations independently.

Looking ahead, this work opens several promising directions for further exploration. In earlier studies [1,2], a new kind of convolution–the t-deformed free convolution, denoted by $\begin{array}{|c|}\hline t\\ \hline\end{array}$–was introduced. For $\rho$, $\eta \in \mathbb{P}$, it is given by

$$\rho \begin{array}{|c|}\hline t\\ \hline\end{array}\eta =U_{1/t}(U_t\left(\rho \right)⊞U_t\left(\eta \right))$$

Inspired by this construction, we observed that for $t>0$ and $a\in \mathbb{R}$, the ${\mathcal{T}}_{(a,t)}$-operator admits an elegant inversion formula:

$${\mathcal{T}}_{(a,t)}^{-1}={\mathcal{T}}_{(-a,1/t)}.$$

In fact, let $\rho \in \mathbb{P}$, $a^{\prime }$, $a^{″}\in \mathbb{R}$ and $t^{\prime }$, $t^{″}>0$ so that

$${\mathcal{T}}_{(a^{\prime },t^{\prime })}\left({\mathcal{T}}_{(a^{″},t^{″})}\left(\rho \right)\right)=\rho .$$

Based on (3), we obtain

$${\displaystyle \frac{1}{{\mathbf{G}}_{\rho }\left(w\right)}}={\displaystyle \frac{1}{{\mathbf{G}}_{{\mathcal{T}}_{(a^{\prime },t^{\prime })}\left({\mathcal{T}}_{(a^{″},t^{″})}\left(\rho \right)\right)}\left(w\right)}}={\displaystyle \frac{t^{\prime }{t}^{″}}{{\mathbf{G}}_{\rho }(w-a^{\prime }-a^{″})}}+t^{\prime }(1-t^{″})(w-a^{\prime })+a^{″}{t}^{″}{t}^{\prime }+(1-t^{\prime })w+a^{\prime }{t}^{\prime }.$$

This implies that $a^{″}=-a^{\prime }$ and $t^{″}=1/t^{\prime }$.

This demonstrates that the family of operators $\left\{{\mathcal{T}}_{(a,t)}\right\}$ forms a group under composition, with a clean and symmetric inversion rule that mirrors the analytic structure of the CST.

Building on this observation, we propose a novel kind of deformed free convolution, called the ${\mathcal{T}}_{(a,t)}$-transformed free convolution, and denoted by ${⊞}_{{\mathcal{T}}_{(a,t)}}$. For $\rho ,\eta \in \mathbb{P}$, it is defined by

$$\rho {⊞}_{{\mathcal{T}}_{(a,t)}}\eta ={\mathcal{T}}_{(-a,1/t)}({\mathcal{T}}_{(a,t)}\left(\rho \right)⊞{\mathcal{T}}_{(a,t)}\left(\eta \right)),a\in \mathbb{R}\mathrm{and}t>0.$$

Note that for $a=0$, this convolution reduces to the original $\begin{array}{|c|}\hline t\\ \hline\end{array}$-convolution.

The introduction of the ${⊞}_{{\mathcal{T}}_{(a,t)}}$-operation broadens the theoretical landscape of free probability. It provides a flexible framework capable of capturing a wider class of deformations, offering new opportunities to study limiting distributions, stability phenomena, and algebraic symmetries. We anticipate that further research into this generalized convolution will deepen the unification between different transformation theories and yield fresh insights into random matrix models, operator algebraic structures, and noncommutative stochastic processes.