On the t-Transformation of Free Convolution

Abstract

The study of the stability of measure families under measure transformations, as well as the accompanying limit theorems, is motivated by both fundamental and applied probability theory and dynamical systems. Stability analysis tries to uncover invariant or quasi-invariant measures that describe the long-term behavior of stochastic or deterministic systems. Limit theorems, on the other hand, characterize the asymptotic distributional behavior of successively changed measures, which frequently indicate convergence to fixed points or attractors. Together, these studies advance our knowledge of measure development, aid in the categorization of dynamical behavior, and give tools for modeling complicated systems in mathematics and applied sciences. In this paper, the notion of the t-transformation of a measure and convolution is studied from the perspective of families and their relative variance functions (VFs). Using analytical and algebraic approaches, we aim to develop a deeper understanding of how the t-transformation shapes the behavior of probability measures, with possible implications in current probabilistic models. Based on the VF concept, we show that the free Meixner family ($\mathcal{FMF}$) of probability measures (the free equivalent of the Letac Mora class) remains invariant when t-transformation is applied. We also use the VFs to show some new limiting theorems concerning t-deformed free convolution and the combination of free and Boolean additive convolution.

1. Introduction

The study of transformations of probability measures and their convolutions is a fundamental topic in probability theory, with broad applications in statistics, stochastic processes, functional analysis, and mathematical physics. Transformations of probability measures, such as those induced by integral transforms, scaling, or functional mappings, provide essential tools for analyzing distributional properties, stability, and limit theorems. Similarly, convolution operations, both classical and generalized, serve as key mechanisms for modeling the aggregation of independent or interacting random variables, playing a crucial role in probability distribution theory. Understanding how probability measures evolve under transformations and convolutions is essential for characterizing their structural properties, such as moment behavior, smoothness, and asymptotic behavior. These operations are central to classical probability, free probability, and other non-commutative frameworks, influencing diverse fields such as random matrix theory, statistical physics, and signal processing.

The concept of measure transformations has been researched and expanded in many ways throughout multiple publications, as presented in [1,2,3]. This article discusses the notion of t-transformation, which was presented in [4,5]. The theoretical basis of t-transformation of probability measures and convolution is examined in relation to the Cauchy–Stieltjes Kernel (CSK) families of measures in free probability. The goal of this work will be discussed in further depth after presenting the central object of this study, the t-transformation.

Denote by $\mathbb{P}$ the set of probabilities on the real line. The Cauchy–Stieltjes transform ${\mathcal{G}}_{\sigma }(·)$ of $\sigma \in \mathbb{P}$ is an integral transform used primarily in complex analysis and probability theory [6]. It is introduced by

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

This transform is useful for studying the distribution of eigenvalues in random matrix theory and recovering the original measure $\sigma$ through its inverse transform under certain conditions. The Cauchy transform of $\sigma$ is subjected to a deformation in the formulation of the t-transformation of measures: Let $t>0$; by the Nevanlinna theorem, the function ${\mathcal{G}}_{{\sigma }_t}(·)$ defined by

$${\displaystyle \frac{1}{{\mathcal{G}}_{{\sigma }_t}\left(\zeta \right)}}={\displaystyle \frac{t}{{\mathcal{G}}_{\sigma }\left(\zeta \right)}}+(1-t)\zeta ,$$

is the Cauchy transform of a certain probability measure denoted by $U_t\left(\sigma \right):={\sigma }_t.$ The t-th Boolean additive convolution power of $\sigma$ is the t-transformation of $\sigma$. The Cauchy transform’s continuous fraction representation may be used to explain the t-deformation of any measure $\sigma$ (with finite moments of all orders). That is, for

$${\displaystyle {\mathcal{G}}_{\sigma }\left(\zeta \right)=\frac{1}{{\displaystyle \zeta -{\kappa }_1-\frac{{\upsilon }_1}{{\displaystyle \zeta -{\kappa }_2-\frac{{\upsilon }_2}{{\displaystyle \zeta -{\kappa }_3-\frac{{\upsilon }_3}{\zeta -{\kappa }_4-\dots }}}}}}},}$$

we have

$${\displaystyle {\mathcal{G}}_{U_t\left(\sigma \right)}\left(\zeta \right)=\frac{1}{{\displaystyle \zeta -t{\kappa }_1-\frac{t{\upsilon }_1}{{\displaystyle \zeta -{\kappa }_2-\frac{{\upsilon }_2}{{\displaystyle \zeta -{\kappa }_3-\frac{{\upsilon }_3}{\zeta -{\kappa }_4-\dots }}}}}}}.}$$

Using t-transformation of measures, a deformation of the free additive convolution ⊞ is considered in [4,5]. It is denoted $\begin{array}{|c|}\hline t\\ \hline\end{array}$ and defined, for ${\nu }_1$ and ${\nu }_2\in \mathbb{P}$, by

$${\nu }_1\begin{array}{|c|}\hline t\\ \hline\end{array}{\nu }_2=U_{1/t}(U_t\left({\nu }_1\right)⊞U_t\left({\nu }_2\right)).$$

Bożejko and Wysoczański [5] explored the relationships between moments and cumulants of a measure and its t-transformation, providing insights into the structural changes induced by the transformation. They computed central limit and Poisson-type limit measures for both transformed classical and free convolutions, demonstrating how these limits depend on the deformation parameter t. Additionally, they constructed families of non-commutative random variables associated with these limit measures, offering examples of “position operators” that act on interacting Fock spaces. This work contributes to the understanding of interpolations between different probabilistic convolutions and their implications in non-commutative probability theory.

This paper aims to advance the understanding of the t-transformation introduced by Bożejko and Wysoczański. However, the study of the stability of measure families under t-transformation, along with the associated limit theorems, plays an important role in non-commutative probability theory. One of the key advantages of analyzing t-transformation is its ability to interpolate between classical and free convolution, thereby offering a unified framework for studying a wide range of probabilistic structures. This approach facilitates the understanding of measure evolution under deformed convolutions and helps identify invariant or limit measures that describe equilibrium or universal behavior in non-commutative settings. However, the theory also presents limitations, including technical difficulties in explicitly characterizing stable families of measures and proving convergence results due to the non-linear and non-classical nature of the t-transformation. Moreover, the lack of intuitive probabilistic interpretations for intermediate t-values can restrict its accessibility. Despite these challenges, the framework offers deep insights into the interplay between different notions of independence and convolution. To address these limitations, the CSK of probability measures defined via analytic transforms, such as the Cauchy–Stieltjes (or resolvent) transform, offer an alternative and more tractable framework. These families are often characterized by their variance functions (VFs), which encode essential information about the measure and its convolutional behavior. The use of VFs allows for an explicit description of the family’s structure and stability properties, simplifying the analysis of evolution under t-transformations. Moreover, the analytic nature of these families provides stronger tools for proving limit theorems and identifying attractors, thereby overcoming many of the difficulties associated with the original formulation of the t-transformation.

The idea of the t-transformation of a measure and of a convolution is examined in this work from the perspective of CSK families and VFs. We aim to develop a deeper understanding of the interaction between t-transformation and the structural features of probability measures by examining their analytic and algebraic qualities. In Section 2, we provide some basic information on the idea of VF of a CSK family in order to make the results to be described more understandable. This idea will be crucial in demonstrating the article’s findings. In Section 3, we introduce some characteristics of the free Letac Mora class and the free Meixner family ($\mathcal{FMF}$) of probability measures. We demonstrate that these two kinds of probability measures remain invariant when performing t-transformation, based on the idea of VF. In Section 4, several new limit theorems related to $\begin{array}{|c|}\hline t\\ \hline\end{array}$-convolution and including free and Boolean additive convolutions are established using the VF notion.

2. The Notion of VF

Similar to the notion of natural exponential families (NEFs) in classical probability, the CSK families in free probability are a novel idea that arises from using the Cauchy–Stieltjes kernel $1/(1-\theta s)$ instead of the exponential kernel $\exp \left(\theta s\right)$. The authors of [7,8] look at compactly supported CSK families. This methodology was expanded in [9] to incorporate probabilities with support constrained from one side, such as from above. The CSK families of probability measures play a crucial role in free probability and free harmonic analysis by providing a powerful analytical framework for studying non-commutative random variables. These families are closely linked to the Stieltjes transform, which serves as a fundamental tool for characterizing free convolution, computing spectral distributions of large random matrices, and analyzing free multiplicative and additive processes. In free harmonic analysis, CSK families facilitate the study of functional transforms, such as the R-transform and subordination functions, which are essential for understanding free convolution and operator-valued distributions. Their structural properties allow for a deeper investigation into asymptotic behavior, stability under transformations, and singularity analysis of free probability measures. By exploring CSK families, we gain significant insights into spectral properties, moment structures, and limiting distributions in free probability, contributing to advancements in random matrix theory, operator algebras, and non-commutative functional analysis.

${\mathcal{P}}_{ba}$ (or ${\mathcal{P}}_c$) represents the set of non-degenerate probabilities with a one-sided support boundary from above (or compact support). Let $\varrho \in {\mathcal{P}}_{ba}$. Then,

$${\mathcal{M}}_{\varrho }\left(\theta \right)=\int {\displaystyle \frac{\varrho \left(ds\right)}{1-\theta s}}$$

is defined for $0\le \theta <{\theta }_{+}^{\varrho }$, where $1/{\theta }_{+}^{\varrho }=\max \{0,\sup \mathrm{supp}\left(\varrho \right)\}$. The set

$${\mathcal{K}}_{+}\left(\varrho \right)=\left\{P_{\varrho }^{\theta }\left(ds\right)={\displaystyle \frac{\varrho \left(ds\right)}{{\mathcal{M}}_{\varrho }\left(\theta \right)(1-\theta s)}}: \theta \in (0,{\theta }_{+}^{\varrho })\right\}$$

is the CSK family generated by $\varrho$.

The mean function $\theta \mapsto k_{\varrho }\left(\theta \right)=\int s{P}_{\varrho }^{\theta }\left(ds\right)$ is bijective from $(0,{\theta }_{+}^{\varrho })$ into its image $(m_0^{\varrho },m_{+}^{\varrho })=k_{\varrho }\left((0,{\theta }_{+}^{\varrho })\right)$, which is the (one sided) mean domain of ${\mathcal{K}}_{+}\left(\varrho \right)$ [9]. This results in a re-parametrization (by the mean) of ${\mathcal{K}}_{+}\left(\varrho \right)$: The inverse of $k_{\varrho }(·)$ is denoted by ${\psi }_{\varrho }(·)$. For $v\in (m_0^{\varrho },m_{+}^{\varrho })$, we write $Q_v^{\varrho }\left(ds\right)=P_{\varrho }^{{\psi }_{\varrho }\left(v\right)}\left(ds\right)$, and we obtain ${\mathcal{K}}_{+}\left(\varrho \right)=\{Q_v^{\varrho }\left(ds\right): v\in (m_0^{\varrho },m_{+}^{\varrho })\}$. We have $m_0^{\varrho }={\lim }_{\theta \to 0^{+}}{k}_{\varrho }\left(\theta \right)$ and $m_{+}^{\varrho }=B_{\varrho }-{\lim }_{z\to B_{\varrho }^{+}}1/{\mathcal{G}}_{\varrho }\left(z\right),$ where $B_{\varrho }=1/{\theta }_{+}^{\varrho }.$

If the support of $\varrho$ is constrained from below, then ${\mathcal{K}}_{-}\left(\varrho \right)$ represents the CSK family. We have ${\theta }_{-}^{\varrho }<\theta <0$, where ${\theta }_{-}^{\varrho }$ is equal to $1/A_{\varrho }$ or $-\infty$ with $A_{\varrho }=\min \{0,\inf \mathrm{supp}\left(\varrho \right)\}$. The mean domain for ${\mathcal{K}}_{-}\left(\varrho \right)$ is $(m_{-}^{\varrho },m_0^{\varrho })$ with $m_{-}^{\varrho }=A_{\varrho }-1/{\mathcal{G}}_{\varrho }\left(A_{\varrho }\right)$. If $\varrho \in {\mathcal{P}}_c$, then ${\theta }_{-}^{\varrho }<\theta <{\theta }_{+}^{\varrho }$, and $\mathcal{K}\left(\varrho \right)={\mathcal{K}}_{-}\left(\varrho \right)\cup \left\{\varrho \right\}\cup {\mathcal{K}}_{+}\left(\varrho \right)$ represent the two-sided CSK family.

Let $\varrho \in {\mathcal{P}}_{ba}$. The VF of ${\mathcal{K}}_{+}\left(\varrho \right)$ is [7]

$$v\mapsto {\mathcal{V}}_{\varrho }\left(v\right)=\int {(s-v)}^2{Q}_v^{\varrho }\left(ds\right).$$

All members of ${\mathcal{K}}_{+}\left(\varrho \right)$ have infinite variance when $\varrho$ does not have the first moment. In [9], a novel definition of the pseudo-variance function (PVF) ${\mathbb{V}}_{\varrho }(·)$ is given as follows: ${\mathbb{V}}_{\varrho }\left(v\right)=v\left(1/{\psi }_{\varrho }\left(v\right)-v\right).$ The VF exists if $m_0^{\varrho }=\int s\varrho \left(ds\right)$ is finite [9], and we have

$${\mathbb{V}}_{\varrho }\left(v\right)={\displaystyle \frac{v{\mathcal{V}}_{\varrho }\left(v\right)}{v-m_0^{\varrho }}}.$$

This section concludes with the following statement, which offers some useful information that supports the proofs of the article’s findings.

Remark 1.

(i) 

A measure $\varrho \in {\mathcal{P}}_{ba}$ is determined by ${\mathbb{V}}_{\varrho }(·)$: For $\Sigma =\Sigma \left(v\right)=v+{\displaystyle \frac{{\mathbb{V}}_{\varrho }\left(v\right)}{v}},$ we have

$${\mathcal{G}}_{\varrho }(\Sigma )={\displaystyle \frac{v}{{\mathbb{V}}_{\varrho }\left(v\right)}}.$$

(1)

If $m_0^{\varrho }$ is exists finitely, then

$${\mathcal{G}}_{\varrho }(\Sigma )={\displaystyle \frac{v-m_0^{\varrho }}{{\mathcal{V}}_{\varrho }\left(v\right)}}.$$

(2)

Thus, ϱ is determined by ${\mathcal{V}}_{\varrho }(·)$ and $m_0^{\varrho }$.

(ii) 

Let $f:s\mapsto \zeta s+\gamma$, where $\gamma \in \mathbb{R}$ and $\zeta \ne 0$. Thus, for v to be near enough to $m_0^{f\left(\varrho \right)}=f\left(m_0^{\varrho }\right)=\zeta m_0^{\varrho }+\gamma$,

$${\mathbb{V}}_{f\left(\varrho \right)}\left(v\right)={\displaystyle \frac{{\zeta }^{2}v}{v-\gamma }}{\mathbb{V}}_{\varrho }\left({\displaystyle \frac{v-\gamma }{\zeta }}\right).$$

(3)

The existence of ${\mathcal{V}}_{\varrho }(·)$ gives

$${\mathcal{V}}_{f\left(\varrho \right)}\left(v\right)={\zeta }^2{\mathcal{V}}_{\varrho }\left({\displaystyle \frac{v-\gamma }{\zeta }}\right).$$

(4)

(iii) 

Let $\varrho \in {\mathcal{P}}_{ba}$. The free cumulant transformation ${\mathcal{R}}_{\varrho }$, of ϱ is defined by [10]

$${\mathcal{R}}_{\varrho }\left({\mathcal{G}}_{\varrho }\left(z\right)\right)=z-1/{\mathcal{G}}_{\varrho }\left(z\right), z closeto \infty .$$

The t-deformation of the free cumulant transformation, denoted by ${\mathcal{R}}^t(·)$, is defined in [4,5] as

$${\mathcal{R}}_{\varrho }^t\left(w\right):={\displaystyle \frac{1}{t}}{\mathcal{R}}_{U_t\left(\varrho \right)}\left(w\right),$$

and for $\lambda >0$, so that ${\varrho }^{\begin{array}{|c|}\hline t\\ \hline\end{array}\lambda }$ is well defined, we have

$${\mathcal{R}}_{{\varrho }^{\begin{array}{|c|}\hline t\\ \hline\end{array}\lambda }}^t\left(w\right)=\lambda {\mathcal{R}}_{\varrho }^t\left(w\right).$$

Furthermore, from [11], we know that for v close enough to $m_0^{U_t\left(\varrho \right)}=t{m}_0^{\varrho }$,

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

(5)

If $m_0^{\varrho }$ is finite,

$${\mathcal{V}}_{U_t\left(\varrho \right)}\left(v\right)=t{\mathcal{V}}_{\varrho }(v/t)+v(v-t{m}_0^{\varrho })(1/t-1).$$

(6)

In addition, from [12] (Corollary 1), one may see that for v near enough to $m_0^{{\varrho }^{\begin{array}{|c|}\hline t\\ \hline\end{array}\lambda }}=\lambda m_0^{\varrho }$,

$${\mathbb{V}}_{{\varrho }^{\begin{array}{|c|}\hline t\\ \hline\end{array}\lambda }}\left(v\right)=\lambda {\mathbb{V}}_{\varrho }(v/\lambda )+v^2((1-t)/\lambda +t-1).$$

(7)

If $m_0^{\varrho }$ is finite,

$${\mathcal{V}}_{{\varrho }^{\begin{array}{|c|}\hline t\\ \hline\end{array}\lambda }}\left(v\right)=\lambda {\mathcal{V}}_{\varrho }(v/\lambda )+v(v-\lambda m_0^{\varrho })((1-t)/\lambda +t-1).$$

(8)

3. Stability Study of the $\mathcal{FMF}$ Under T-Transformation

The class of quadratic CSK families with VF

$${\mathcal{V}}_{\varrho }\left(v\right)=1+av+b{v}^2, a\in \mathbb{R}, b\ge -1, m_0^{\varrho }=0,$$

(9)

were characterized in [7]. The associated measures belong to the $\mathcal{FMF}$:

$$\varrho \left(dl\right)={\displaystyle \frac{\sqrt{4(b+1)-{(l-a)}^2}}{2\pi (b{l}^2+al+1)}}{\mathbf{1}}_{(a-2\sqrt{b+1},a+2\sqrt{b+1})}\left(l\right)dl+m_1{\delta }_{l_1}+m_2{\delta }_{l_2}.$$

We haveIf

(i)

If $a^2>1, b=0$, then $m_1=1-1/a^2, l_1=-1/a, m_2=0$.

(ii)

If $b>0$ and $a^2>4b$, then $m_1=\max \{0,1-{\displaystyle \frac{\left|a\right|-\sqrt{a^2-4b}}{2b\sqrt{a^2-4b}}}\}, m_2=0$, and $l_1=\pm {\displaystyle \frac{\left|a\right|-\sqrt{a^2-4b}}{2b}}$ with the sign opposite to the sign of a.

(iii)

If $-1\le b<0$,

$$l_{1,2}={\displaystyle \frac{-a\pm \sqrt{a^2-4b}}{2b}}, m_{1,2}=1+{\displaystyle \frac{\sqrt{a^2-4b}\mp a}{2b\sqrt{a^2-4b}}}$$

This result includes crucial measures. Up to dilation and convolution, $\varrho$ is

(i)

The semicircle (SC) law if $a=b=0$.

(ii)

The Marchenko–Pastur (MP) law if $b=0$ and $a\ne 0$.

(iii)

The free Pascal (FP) law if $b>0$ and $a^2>4b$.

(iv)

The free Gamma (FG) law if $b>0$ and $a^2=4b$.

(v)

The free analog of hyperbolic (FH) law if $b>0$ and $a^2<4b$.

(vi)

The free binomial (FB) law if $-1\le b<0$.

Next, we demonstrate that when the t-transformation of measures is applied, the $\mathcal{FMF}$ remains invariant. $D_c\left(\rho \right)$ will denote the dilation of a measure $\rho$ by a number $c\ne 0$.

Theorem 1.

If $\rho \in \mathcal{FMF}$, then for each $t>0$, $U_{1/t}\left(D_{\sqrt{t}}\left(\rho \right)\right)\in \mathcal{FMF}$.

Proof. 

Suppose that $\rho \in \mathcal{FMF}$. Then, for $\mathcal{K}\left(\rho \right)$, the VF may therefore be expressed as

$${\mathcal{V}}_{\rho }\left(v\right)=1+a^{\prime }v+b^{\prime }{v}^2, a^{\prime }\in \mathbb{R}, b^{\prime }\ge -1.$$

(10)

Combining (10), (6) and (4), for v close to $m_0^{U_{1/t}\left(D_{\sqrt{t}}\left(\rho \right)\right)}=0$, one obtains

$${\mathcal{V}}_{U_{1/t}\left(D_{\sqrt{t}}\left(\rho \right)\right)}\left(v\right)=1+a^{\prime }\sqrt{t}v+(b^{\prime }t+t-1)v^2,$$

(11)

which is a VF of the type (9). Then, $U_{1/t}\left(D_{\sqrt{t}}\left(\rho \right)\right)\in \mathcal{FMF}$.    □

This result shows that the $\mathcal{FMF}$ is closed under t-transformation, which is a non-trivial algebraic and analytical property. Closure under such a transformation provides a strong reason to study and classify these measures further, as it reflects their centrality in both theoretical and applied contexts in non-commutative probability. In the following, we demonstrate the significance of Theorem 1 using specific measures.

Corollary 1.

Consider $sb\left(dl\right)={\displaystyle \frac{1}{2}}{\delta }_{-1}+{\displaystyle \frac{1}{2}}{\delta }_1$. Then, $U_{1/t}\left(D_{\sqrt{t}}\left(sb\right)\right)=sb$.

Proof. 

The VF of $\mathcal{K}\left(sb\right)$ is provided by (10) for $a^{\prime }=0$ and $b^{\prime }=-1$. From (11), we have that ${\mathcal{V}}_{U_{1/t}\left(D_{\sqrt{t}}\left(sb\right)\right)}\left(v\right)=1-v^2={\mathcal{V}}_{sb}\left(v\right)$. This concludes the proof using (2).   □

Corollary 2.

Consider the SC law

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

(12)

Then, $U_{1/t}\left(D_{\sqrt{t}}\left(sc\right)\right)$ is

(i) 

A SC law if $t=1$.

(ii) 

A FH law with $a=0$ and $b=t-1>0$ if $t\in (1,+\infty )$.

(iii) 

A FB law with $a=0$ and $-1\le b=t-1<0$ if $t\in (0,1)$.

Proof. 

For $\mathcal{K}\left(sc\right)$, the VF provided by (10) with $a^{\prime }=b^{\prime }=0$. From (11), we have that

$${\mathcal{V}}_{U_{1/t}\left(D_{\sqrt{t}}\left(sc\right)\right)}\left(m\right)=1+(t-1)m^2,$$

(13)

An identification between (13) and (9) gives the following:

(i)

If $t=1$, then $U_{1/t}\left(D_{\sqrt{t}}\left(sc\right)\right)=sc$.

(ii)

If $t\in (1,+\infty )$, then $U_{1/t}\left(D_{\sqrt{t}}\left(sc\right)\right)$ is an FH law with $a=0$ and $b=t-1>0$.

(iii)

If $t\in (0,1)$, then $U_{1/t}\left(D_{\sqrt{t}}\left(sc\right)\right)$ is a FB law with $a=0$ and $-1\le b=t-1<0$.

□

Corollary 3.

For $a^{\prime }\ne 0$ and $b^{\prime }=0$, consider the MP law

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

(14)

Then, $U_{1/t}\left(D_{\sqrt{t}}\left(mp\right)\right)$ is

(i) 

An MP law with $a=a^{\prime }$ and $b=b^{\prime }=0$ if $t=1$.

(ii) 

An FP law with $a=a^{\prime }\sqrt{t}$ and $b=t-1>0$ if $t>1$ and $t{a}^{\prime 2}>4(t-1)$.

(iii) 

An FG law with $a=a^{\prime }\sqrt{t}$ and $b=t-1>0$ if $t>1$ and $t{a}^{\prime 2}=4(t-1)$.

(iv) 

An FH law with $a=a^{\prime }\sqrt{t}$ and $b=t-1>0$ if $t>1$ and $t{a}^{\prime 2}<4(t-1)$.

(v) 

An FB law with $a=a^{\prime }\sqrt{t}$ and $-1\le b=t-1<0$ if $t\in (0,1)$.

Proof. 

We have $a^{\prime }\ne 0$ and $b^{\prime }=0$ for the VF of $\mathcal{K}\left(mp\right)$ provided by (10). From (11), one obtains

$${\mathcal{V}}_{U_{1/t}\left(D_{\sqrt{t}}\left(mp\right)\right)}\left(v\right)=1+a^{\prime }\sqrt{t}v+(t-1)v^2.$$

(15)

An identification between (15) and (9) gives the following:

(i)

If $t=1$, then $U_{1/t}\left(D_{\sqrt{t}}\left(mp\right)\right)=mp$.

(ii)

If $t>1$ and $t{a}^{\prime 2}>4(t-1)$, then $U_{1/t}\left(D_{\sqrt{t}}\left(mp\right)\right)$ is an FP law with $a=a^{\prime }\sqrt{t}$ and $b=t-1>0$.

(iii)

If $t>1$ and $t{a}^{\prime 2}=4(t-1)$, then $U_{1/t}\left(D_{\sqrt{t}}\left(mp\right)\right)$ is an FG law with $a=a^{\prime }\sqrt{t}$ and $b=t-1>0$.

(iv)

If $t>1$ and $t{a}^{\prime 2}<4(t-1)$, then $U_{1/t}\left(D_{\sqrt{t}}\left(mp\right)\right)$ is an FH law with $a=a^{\prime }\sqrt{t}$ and $b=t-1>0$.

(v)

If $t\in (0,1)$, then $U_{1/t}\left(D_{\sqrt{t}}\left(mp\right)\right)$ is an FB law with $a=a^{\prime }\sqrt{t}$ and $-1\le b=t-1<0$.

□

Corollary 4.

For $a^{\prime }\ne 0$ and $b^{\prime }=a^{\prime 2}/4$, consider the FG law

$$fg\left(dl\right)={\displaystyle \frac{\sqrt{4(1+b^{\prime })-{(l-a^{\prime })}^2}}{2\pi (b^{\prime }{l}^2+a^{\prime }l+1)}}{\mathbf{1}}_{(a^{\prime }-2\sqrt{1+b^{\prime }},a^{\prime }+2\sqrt{1+b^{\prime }})}\left(l\right)dl.$$

(16)

Then, $U_{1/t}\left(D_{\sqrt{t}}\left(fg\right)\right)$ is

(i) 

An FG law with $a=a^{\prime }$ and $b=b^{\prime }$ if $t=1$.

(ii) 

An FH law with $a=a^{\prime }\sqrt{t}$ and $b=b^{\prime }t+t-1=a^{\prime 2}t/4+t-1>0$ if $t\in (1,+\infty )$.

(iii) 

An FP law with $a=a^{\prime }\sqrt{t}$ and $b=b^{\prime }t+t-1=a^{\prime 2}t/4+t-1$ if $t\in (0,1)$ such that $a^{\prime 2}t+4(t-1)>0$.

(iv) 

An FB law with $a=a^{\prime }\sqrt{t}$ and $b=b^{\prime }t+t-1=a^{\prime 2}t/4+t-1$ if $t\in (0,1)$, such that $-1\le a^{\prime 2}t/4+t-1<0$.

Proof. 

We have $a^{\prime }\ne 0$ and $b^{\prime }=a^{\prime 2}/4$ for the VF of $\mathcal{K}\left(fg\right)$ given by (10). From (11), we have that

$${\mathcal{V}}_{U_{1/t}\left(D_{\sqrt{t}}\left(fg\right)\right)}\left(v\right)=1+a^{\prime }\sqrt{t}v+(b^{\prime }t+t-1)v^2=1+a^{\prime }\sqrt{t}v+(a^{\prime 2}t/4+t-1)v^2.$$

(17)

An identification between (17) and (9) gives the following:

(i)

If $t=1$, then $U_{1/t}\left(D_{\sqrt{t}}\left(fg\right)\right)=fg$.

(ii)

If $t\in (1,+\infty )$, then $U_{1/t}\left(D_{\sqrt{t}}\left(fg\right)\right)$ is an FH law with $a=a^{\prime }\sqrt{t}$ and $b=b^{\prime }t+t-1=a^{\prime 2}t/4+t-1$.

(iii)

If $t\in (0,1)$ such that $a^{\prime 2}t+4(t-1)>0$, then $U_{1/t}\left(D_{\sqrt{t}}\left(fg\right)\right)$ is an FP law with $a=a^{\prime }\sqrt{t}$ and $b=b^{\prime }t+t-1=a^{\prime 2}t/4+t-1>0$.

(iv)

If $t\in (0,1)$, such that $-1\le a^{\prime 2}t/4+t-1<0$, then $U_{1/t}\left(D_{\sqrt{t}}\left(fg\right)\right)$ is an FB law with $a=a^{\prime }\sqrt{t}$ and $b=b^{\prime }t+t-1=a^{\prime 2}t/4+t-1$.

□

Because free Meixner distributions arise in the context of creation and annihilation operators on Fock spaces, their invariance under t-transformation has implications for operator models and quantum probability, where understanding deformations of independence is crucial. Next, we show that the $\mathcal{FMF}$ is also invariant when considering the $\begin{array}{|c|}\hline t\\ \hline\end{array}$-convolution power.

Theorem 2.

If $\rho \in \mathcal{FMF}$, then $D_{1/\sqrt{\alpha }}\left({\rho }^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)\in \mathcal{FMF}$ for $\alpha >0$ so that ${\rho }^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }$ is defined.

Proof. 

Assume that $\rho \in \mathcal{FMF}$. Combining (10), (8) and (4), for v close enough to $m_0^{D_{1/\sqrt{\alpha }}\left({\rho }^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)}=0$, one obtains

$${\mathcal{V}}_{D_{1/\sqrt{\alpha }}\left({\rho }^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)}\left(v\right)=1+{\displaystyle \frac{a^{\prime }}{\sqrt{\alpha }}}v+\left({\displaystyle \frac{b^{\prime }+1-t}{\alpha }}+t-1\right)v^2,$$

(18)

which is a VF of the type (9). Then, $D_{1/\sqrt{\alpha }}\left({\rho }^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)\in \mathcal{FMF}$.    □

The invariance of the $\mathcal{FMF}$ under the t-transformation highlights their foundational role in free and interpolated probability theories. It points to deep algebraic and analytical properties that make these measures central objects of study. Next, the significance of Theorem 2 is emphasized throughout a few specific measures.

Corollary 5.

Consider $sb\left(dl\right)={\displaystyle \frac{1}{2}}{\delta }_{-1}+{\displaystyle \frac{1}{2}}{\delta }_1$. Then $D_{1/\sqrt{\alpha }}\left(s{b}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)$ is

(i) 

A symmetric Bernoulli measure if $\alpha =1$.

(ii) 

An SC law if $\alpha \in (1,+\infty )$ and $t={\displaystyle \frac{1}{1-1/\alpha }}$.

(iii) 

An FH law with $a=0$ and $b={\displaystyle \frac{-t}{\alpha }}+t-1$ if $\alpha \in (1,+\infty )$ and $t>{\displaystyle \frac{1}{1-1/\alpha }}$.

(iv) 

An FB law with $a=0$ and $b={\displaystyle \frac{-t}{\alpha }}+t-1$ if $\alpha \in (1,+\infty )$ and $t<{\displaystyle \frac{1}{1-1/\alpha }}$.

Proof. 

The VF of $\mathcal{K}\left(sb\right)$ is provided by (10) for $a^{\prime }=0$ and $b^{\prime }=-1$. From (18), one obtains

$${\mathcal{V}}_{D_{1/\sqrt{\alpha }}\left(s{b}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)}\left(v\right)=1+\left(-t/\alpha +t-1\right)v^2.$$

(19)

The rest of the proof follows by comparing (19) to (9).    □

Corollary 6.

Consider the SC law (12). Then, $D_{1/\sqrt{\alpha }}\left(s{c}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)$ is

(i) 

An SC law if $\alpha =1$ or $t=1$.

(ii) 

An FH law with $a=0$ and $b=(1-t)(1/\alpha -1)$ if $\alpha \in (1,+\infty )$ and $t\in (1,+\infty )$ (or $\alpha \in (0,1)$ and $t\in (0,1)$).

(iii) 

An FB law with $a=0$ and $-1\le b=(1-t)(1/\alpha -1)<0$ if $\alpha \in (1,+\infty )$ and $t\in (0,1)$ (or $\alpha \in (0,1)$ and $t\in (1,+\infty )$ such that $(1-t)(1/\alpha -1)\ge -1$).

Proof. 

Consider the SC law given by (12); then, the relation (18) is

$${\mathcal{V}}_{D_{1/\sqrt{\alpha }}\left(s{c}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)}\left(v\right)=1+(1-t)\left(1/\alpha -1\right)v^2.$$

(20)

The rest of the proof follows by comparing (20) to (9).    □

Corollary 7.

Consider the MP law (14). Then, $D_{1/\sqrt{\alpha }}\left(m{p}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)$ is

(i) 

An MP law with $a=a^{\prime }/\sqrt{\alpha }$ and $b=b^{\prime }=0$ if $\alpha =1$ or $t=1$.

(ii) 

An FP law with $a=a^{\prime }/\sqrt{\alpha }$ and $b=(1-t)(1/\alpha -1)$ if $\alpha \in (0,1)$ and $t\in (0,1)$ (or $\alpha \in (1,+\infty )$ and $t\in (1,+\infty )$) such that $a^{\prime 2}/\alpha >4(1-t)(1/\alpha -1)$.

(iii) 

An FG law with $a=a^{\prime }/\sqrt{\alpha }$ and $b=(1-t)(1/\alpha -1)$ if $\alpha \in (0,1)$ and $t\in (0,1)$ (or $\alpha \in (1,+\infty )$ and $t\in (1,+\infty )$) such that $a^{\prime 2}/\alpha =4(1-t)(1/\alpha -1)$.

(iv) 

An FH law with $a=a^{\prime }/\sqrt{\alpha }$ and $b=(1-t)(1/\alpha -1)$ if $\alpha \in (0,1)$ and $t\in (0,1)$ (or $\alpha \in (1,+\infty )$ and $t\in (1,+\infty )$) such that $a^{\prime 2}/\alpha <4(1-t)(1/\alpha -1)$.

(v) 

An FB law with $a=a^{\prime }/\sqrt{\alpha }$ and $b=(1-t)(1/\alpha -1)$ if $\alpha \in (1,+\infty )$ and $t\in (0,1)$ (or $\alpha \in (0,1)$ and $t\in (1,+\infty )$ such that $-1\le b=(1-t)(1/\alpha -1)<0$).

Proof. 

Consider the MP law (14). Then, relation (18) reduces to

$${\mathcal{V}}_{D_{1/\sqrt{\alpha }}\left(m{p}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)}\left(v\right)=1+{\displaystyle \frac{a^{\prime }}{\sqrt{\alpha }}}v+(1-t)\left(1/\alpha -1\right)v^2.$$

(21)

The rest of the proof follows by comparing (21) to (9).    □

Corollary 8.

Consider the FG law (16). Then, $D_{1/\sqrt{\alpha }}\left(f{g}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)$ is

(i) 

An FG law with $a=a^{\prime }/\sqrt{\alpha }$ and $b={\displaystyle \frac{a^{\prime 2}/4+1-t}{\alpha }}+t-1$ if $\alpha =1$ or $t=1$.

(ii) 

An FH law with $a=a^{\prime }/\sqrt{\alpha }$ and $b={\displaystyle \frac{a^{\prime 2}/4+1-t}{\alpha }}+t-1$ if $\alpha \in (0,1)$ and $t\in (0,1)$ (or $\alpha \in (1,+\infty )$ and $t\in (1,+\infty )$ such that $a^{\prime 2}/\alpha >4(t-1)(1/\alpha -1))$.

(iii) 

An FP law with $a=a^{\prime }/\sqrt{\alpha }$ and $b={\displaystyle \frac{a^{\prime 2}/4+1-t}{\alpha }}+t-1$ if $\alpha \in (0,1)$ and $t\in (1,+\infty )$ (or $\alpha \in (1,+\infty )$ and $t\in (0,1)$) such that $a^{\prime 2}/\alpha >4(t-1)(1/\alpha -1)$.

(iv) 

An FB law with $a=a^{\prime }/\sqrt{\alpha }$ and $b={\displaystyle \frac{a^{\prime 2}/4+1-t}{\alpha }}+t-1$ if $\alpha \in (1,+\infty )$ and $t\in (0,1)$ (or $\alpha \in (0,1)$ and $t\in (1,+\infty )$) such that $-1\le b={\displaystyle \frac{a^{\prime 2}/4+1-t}{\alpha }}+t-1<0$.

Proof. 

Consider the FG law (16); then, relation (18) reduces to

$${\mathcal{V}}_{D_{1/\sqrt{\alpha }}\left(f{g}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)}\left(v\right)=1+{\displaystyle \frac{a^{\prime }}{\sqrt{\alpha }}}v+\left({\displaystyle \frac{a^{\prime 2}/4+1-t}{\alpha }}+t-1\right)v^2.$$

(22)

The rest of the proof follows by comparing (22) to (9).    □

4. Stability Study Under t-Transformation of CSK Families with Cubic PVF

A class of cubic CSK families with PVF

$${\mathbb{V}}_{\tau }\left(v\right)=v(a{v}^2+bv+c),$$

(23)

where $m_0^{\tau }=-\infty$ and $a>0$ are described in [9]. The associated laws are

$$\tau \left(dl\right)={\displaystyle \frac{\sqrt{4ac-{(b+1)}^2-4al}}{2\pi (c+bl+a{l}^2)}}{\mathbf{1}}_{\left(-\infty ,c-{\displaystyle \frac{{(b+1)}^2}{\left(4a\right)}}\right)}\left(l\right)dl+m(a,b,c){\delta }_{-(b+\sqrt{b^2-4ac})/\left(2a\right)},$$

(24)

where $m(a,b,c)=1-1/\sqrt{b^2-4ac}$ if $b^2>4ac+1$ and is 0 otherwise. The most significant example in this class is the inverse semicircle (ISC) law:

$$isc\left(dl\right)={\displaystyle \frac{p\sqrt{-p^2-4l}}{2\pi l^2}}{\mathbf{1}}_{(-\infty ,-p^2/4)}\left(l\right)dl, p>0.$$

(25)

We have

$${\mathbb{V}}_{isc}\left(v\right)=v^3/p^2, v\in (m_0^{isc},m_{+}^{isc})=(-\infty ,-p^2).$$

(26)

Additionally, we have the free analogous of the five other Letac Mora class members; see [9] (pp. 590–591). The class of measures of the type (24) is denoted by $\mathcal{CPVF}$. We have the following result:

Theorem 3.

If $\tau \in \mathcal{CPVF}$, then for $t>0$, $U_{1/t}\left(D_{\sqrt{t}}\left(\tau \right)\right)\in \mathcal{CPVF}$.

Proof. 

Assume that $\tau \in \mathcal{CPVF}$. Then, for ${\mathcal{K}}_{+}\left(\tau \right)$, the PVF is given by

$${\mathbb{V}}_{\tau }\left(v\right)=v(a^{\prime }{v}^2+b^{\prime }v+c^{\prime }), a^{\prime }>0, b^{\prime }, c^{\prime }\in \mathbb{R}.$$

(27)

Combining (27), (3), and (5), for v close enough to $m_0^{U_{1/t}\left(D_{\sqrt{t}}\left(\tau \right)\right)}=-\infty$, we obtain

$${\mathbb{V}}_{U_{1/t}\left(D_{\sqrt{t}}\left(\tau \right)\right)}\left(v\right)=v\left(a^{\prime }t\sqrt{t}{v}^2+\left(b^{\prime }t+t-1\right)v+c^{\prime }\sqrt{t}\right),$$

(28)

which is a PVF of the form (23). Thus, $U_{1/t}\left(D_{\sqrt{t}}\left(\tau \right)\right)\in \mathcal{CPVF}$.    □

Corollary 9.

Consider the ISC law (25) with $m_0^{isc}=-\infty$ and parameters $a^{\prime }=1/p^2={\displaystyle \frac{1}{2\sqrt{2}}}$, $b^{\prime }=c^{\prime }=0$. Then, $U_{1/2}\left(D_{\sqrt{2}}\left(isc\right)\right)$ is the free Ressel (or free Kendall) law as presented in [9] (Section 4.2(ii)).

Proof. 

Combining (28) and (26), one sees that the PVF of ${\mathcal{K}}_{+}(U_{1/2}(D_{\sqrt{2}}(isc)))$ is given by ${\mathbb{V}}_{U_{1/2}\left(D_{\sqrt{2}}\left(isc\right)\right)}\left(v\right)=v^2(v+1)$, which is the PVF of the free Ressel CSK family; see [9] (Section 4.2(ii)). The proof is concluded by (1).    □

Next, we show that the $\mathcal{CPVF}$ is also invariant when considering $\begin{array}{|c|}\hline t\\ \hline\end{array}$-convolution power.

Theorem 4.

If $\tau \in \mathcal{CPVF}$, then $D_{1/\sqrt{\alpha }}\left({\tau }^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)\in \mathcal{CPVF}$ for $\alpha >0$ so that ${\tau }^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }$ is defined.

Proof. 

Assume that $\tau \in \mathcal{CPVF}$. Using (27), (3) and (7), for v close enough to $m_0^{D_{1/\sqrt{\alpha }}\left({\tau }^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)}=-\infty$, we obtain

$${\mathbb{V}}_{D_{1/\sqrt{\alpha }}\left({\tau }^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)}\left(v\right)=v\left(a^{\prime }{v}^2+\left(b^{\prime }+(\alpha -1)(t-1)\right)v+c^{\prime }\right),$$

(29)

which is a PVF of the form (23). Thus, $D_{1/\sqrt{\alpha }}\left({\tau }^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)\in \mathcal{CPVF}$.    □

Corollary 10.

Consider the ISC law (25) with $m_0^{isc}=-\infty$ and parameters $a^{\prime }=1/p^2=1$, $b^{\prime }=c^{\prime }=0$. Then,

(i) 

For $\alpha \in (0,1)$ and $t=(2-\alpha )/(1-\alpha )$, measure $D_{1/\sqrt{\alpha }}\left({isc}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)$ is a free Abel law of the form (24) with parameters $a=1$, $b=-1$ and $c=0$.

(ii) 

For $\alpha \in (1,+\infty )$ and $t=\alpha /(\alpha -1)$, measure $D_{1/\sqrt{\alpha }}\left({isc}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)$ is a free Ressel (or free Kendall) law of the form (24) with parameters $a=1$, $b=1$ and $c=0$.

Proof. 

Combining (29) and (26) (for $a^{\prime }=1$, $b^{\prime }=c^{\prime }=0$), one sees that

$${\mathbb{V}}_{D_{1/\sqrt{\alpha }}\left({isc}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)}\left(v\right)=v\left(v^2+(\alpha -1)(t-1)v\right).$$

(i)

For $\alpha \in (0,1)$ and $t=(2-\alpha )/(1-\alpha )$, one sees that

$${\mathbb{V}}_{D_{1/\sqrt{\alpha }}\left({isc}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)}\left(v\right)=v^2\left(v-1\right)$$

which is the PVF of the free Abel CSK family; see [9] (Section 4.2(i)). The proof of (i) is concluded by (1).

(ii)

For $\alpha \in (1,+\infty )$ and $t=\alpha /(\alpha -1)$, one sees that

$${\mathbb{V}}_{D_{1/\sqrt{\alpha }}\left({isc}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\right)}\left(v\right)=v^2\left(v+1\right)$$

which is the PVF of the free Ressel CSK family; see [9] (Section 4.2(ii)). The proof of (ii) is concluded by (1).    □

5. Limit Theorems Related to $\begin{array}{|c|}\hline t\\ \hline\end{array}$-Convolution

We introduce novel limiting theorems for the $\begin{array}{|c|}\hline t\\ \hline\end{array}$-convolution, including free and Boolean additive convolutions. The free additive convolution ⊞ is just the $\begin{array}{|c|}\hline t\\ \hline\end{array}$-convolution with $t=1$. The Boolean additive convolution power ${\sigma }^{\uplus t}$ is just the t-transformation of $\sigma$. The results in this section also involve the transformation of measures [13]

$$\begin{array}{ccc}\hfill {\mathbb{B}}_r:\mathcal{P}& \to & \mathcal{P}\hfill \\ \hfill \sigma & \mapsto & {\mathbb{B}}_r\left(\sigma \right)={\left({\sigma }^{⊞(1+r)}\right)}^{\uplus {\displaystyle \frac{1}{1+r}}}, r\ge 0.\hfill \end{array}$$

For $r=1$, the transformation ${\mathbb{B}}_1$ corresponds to Bercovici and Pata bijection.

Let $\sigma \in {\mathcal{P}}_{ba}$ with a finite first moment. Recall from [9] that for $p>0$, so that ${\sigma }^{⊞p}$ is defined, and for x near enough to $m_0^{{\sigma }^{⊞p}}=p{m}_0^{\sigma }$,

$${\mathcal{V}}_{{\sigma }^{⊞p}}\left(x\right)=p{\mathcal{V}}_{\sigma }(x/p).$$

(30)

Furthermore, for $r\ge 0$ and for x close enough to $m_0^{\sigma }$ [11],

$${\mathcal{V}}_{{\mathbb{B}}_r\left(\sigma \right)}\left(x\right)={\mathcal{V}}_{\sigma }\left(x\right)+rx(x-m_0^{\sigma }) \mathrm{and} {\mathcal{V}}_{{\mathbb{B}}_1^{-1}\left(\sigma \right)}\left(x\right)={\mathcal{V}}_{\sigma }\left(x\right)-x(x-m_0^{\sigma }).$$

(31)

Next, we state and show the main results of this paragraph.

Theorem 5.

Let $\rho \in {\mathcal{P}}_c$. Then,

(i) 

For $t\in [0,1]$ and for $\alpha >0$ so that ${\left({\rho }^{\begin{array}{|c|}\hline t\\ \hline\end{array}{\displaystyle \frac{1}{\alpha }}}\right)}^{⊞\alpha }$ is defined,

$${\left({\rho }^{\begin{array}{|c|}\hline t\\ \hline\end{array}{\displaystyle \frac{1}{\alpha }}}\right)}^{⊞\alpha }\stackrel{\alpha \to +\infty }{\to }{\mathbb{B}}_{1-t}\left(\rho \right), indistribution.$$

(32)

(ii) 

For $t\in [1,+\infty )$ and for $\alpha >0$ so that ${\left({\rho }^{⊞{\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }$ is defined,

$${\left({\rho }^{⊞{\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\stackrel{\alpha \to +\infty }{\to }{\mathbb{B}}_{t-1}\left(\rho \right), indistribution.$$

(33)

(iii) 

For $t\ge 0$ and for $\alpha >0$ so that ${\left({\rho }^{\uplus {\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }$ is defined,

$${\left({\rho }^{\uplus {\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\stackrel{\alpha \to +\infty }{\to }{\mathbb{B}}_t\left(\rho \right), indistribution.$$

(34)

Proof. 

(i) Using (30), (31) and (8), for x close enough to $m_0^{{\left({\rho }^{\begin{array}{|c|}\hline t\\ \hline\end{array}{\displaystyle \frac{1}{\alpha }}}\right)}^{⊞\alpha }}=m_0^{\rho }$, we have

$$\begin{array}{ccc}\hfill {\mathcal{V}}_{{\left({\rho }^{\begin{array}{|c|}\hline t\\ \hline\end{array}{\displaystyle \frac{1}{\alpha }}}\right)}^{⊞\alpha }}\left(x\right)& =& \alpha {\mathcal{V}}_{{\rho }^{\begin{array}{|c|}\hline t\\ \hline\end{array}{\displaystyle \frac{1}{\alpha }}}}(x/\alpha )={\mathcal{V}}_{\rho }\left(x\right)+x(x-m_0^{\rho })(1-t+(t-1)/\alpha )\hfill \\ & \stackrel{\alpha \to +\infty }{\to }& {\mathcal{V}}_{\rho }\left(x\right)+(1-t)x(x-m_0^{\rho })={\mathcal{V}}_{{\mathbb{B}}_{1-t}\left(\rho \right)}\left(x\right).\hfill \end{array}$$

The proof of (32) is concluded by the use of [7] (Proposition 4.2).

(ii) Using (30), (31) and (8), for x close enough to $m_0^{{\left({\rho }^{⊞{\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }}=m_0^{\rho }$, we have

$$\begin{array}{ccc}\hfill {\mathcal{V}}_{{\left({\rho }^{⊞{\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }}\left(x\right)& =& \alpha {\mathcal{V}}_{{\rho }^{⊞{\displaystyle \frac{1}{\alpha }}}}(x/\alpha )+x(x-\alpha m_0^{{\rho }^{⊞{\displaystyle \frac{1}{\alpha }}}})(t-1+(1-t)/\alpha )\hfill \\ & =& {\mathcal{V}}_{\rho }\left(x\right)+x(x-m_0^{\rho })(t-1+(1-t)/\alpha )\hfill \\ & \stackrel{\alpha \to +\infty }{\to }& {\mathcal{V}}_{\rho }\left(x\right)+(t-1)x(x-m_0^{\rho })={\mathcal{V}}_{{\mathbb{B}}_{t-1}\left(\rho \right)}\left(x\right).\hfill \end{array}$$

Using [7] (Proposition 4.2), this end the proof of (33).

(iii) Using (6), (31) and (8), for x close enough to $m_0^{{\left({\rho }^{\uplus {\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }}=m_0^{\rho }$,

$$\begin{array}{ccc}\hfill {\mathcal{V}}_{{\left({\rho }^{\uplus {\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }}\left(x\right)& =& \alpha {\mathcal{V}}_{{\rho }^{\uplus {\displaystyle \frac{1}{\alpha }}}}(x/\alpha )+x(x-\alpha m_0^{{\rho }^{\uplus {\displaystyle \frac{1}{\alpha }}}})(t-1+(1-t)/\alpha )\hfill \\ & =& \alpha \left[{\mathcal{V}}_{\rho }\left(x\right)/\alpha +{\displaystyle \frac{x}{\alpha }}(x/\alpha -m_0^{\rho }/\alpha )(\alpha -1)\right]+x(x-m_0^{\rho })(t-1+(1-t)/\alpha )\hfill \\ & =& {\mathcal{V}}_{\rho }\left(x\right)+x(x-m_0^{\rho })(1-1/\alpha )+x(x-m_0^{\rho })(t-1+(1-t)/\alpha )\hfill \\ & \stackrel{\alpha \to +\infty }{\to }& {\mathcal{V}}_{\rho }\left(x\right)+x(x-m_0^{\rho })+x(x-m_0^{\rho })(t-1)={\mathcal{V}}_{{\mathbb{B}}_t\left(\rho \right)}\left(x\right).\hfill \end{array}$$

This, together with [7] (Proposition 4.2) completes the proof of (34).

□

Next, the significance of Theorem 5 is emphasized through a few specific measures. This will offer some interesting insights on the t-deformed free Gaussian law (or Kesten law), which is given by ${\kappa }_t=\widetilde{{\kappa }_t}+\widehat{{\kappa }_t}$ (see [14]) with

$$\widetilde{{\kappa }_t}\left(dl\right)={\displaystyle \frac{1}{2\pi }}{\displaystyle \frac{\sqrt{4t-l^2}}{1-(1-t)l^2}}{\mathbf{1}}_{[-2\sqrt{t},2\sqrt{t}]}\left(l\right)dl$$

and for $t<1/2$

$$\widehat{{\kappa }_t}={\displaystyle \frac{1-2t}{2(1-t)}}\left({\delta }_{-1/\sqrt{1-t}}+{\delta }_{1/\sqrt{1-t}}\right).$$

Example 1.

Consider the SC law (12). Then, for $t\ge 1$, we have

$${\left(s{c}^{⊞{\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\stackrel{\alpha \to +\infty }{\to }{\mathbb{B}}_{t-1}\left(sc\right)={\kappa }_t, indistribution.$$

In fact, for x close enough to $m_0^{{\left(s{c}^{⊞{\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }}=m_0^{sc}=0$, we have

$${\mathcal{V}}_{{\left(s{c}^{⊞{\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }}\left(x\right)\stackrel{\alpha \to +\infty }{\to }{\mathcal{V}}_{sc}\left(x\right)+(t-1)x(x-m_0^{sc})=1+(t-1)x^2={\mathcal{V}}_{{\kappa }_t}\left(x\right).$$

See [12] (Proposition 3) for the calculation of ${\mathcal{V}}_{{\kappa }_t}(·)$.

Example 2.

Consider $sb\left(dl\right)={\displaystyle \frac{1}{2}}{\delta }_{-1}+{\displaystyle \frac{1}{2}}{\delta }_1$. Then, for $t\ge 0$, we have

$${\left(s{b}^{\uplus {\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }\stackrel{\alpha \to +\infty }{\to }{\mathbb{B}}_t\left(sb\right)={\kappa }_t, indistribution.$$

In fact, for x close enough to $m_0^{{\left(s{b}^{\uplus {\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }}=m_0^{sb}=0$, we have

$${\mathcal{V}}_{{\left(s{b}^{\uplus {\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline t\\ \hline\end{array}\alpha }}\left(x\right)\stackrel{\alpha \to +\infty }{\to }{\mathcal{V}}_{sb}\left(x\right)+tx(x-m_0^{sb})=1-x^2+t{x}^2={\mathcal{V}}_{{\kappa }_t}\left(x\right).$$

Example 3.

Consider the Kesten law ${\kappa }_t$. Then, for $t\in [0,1]$, we have

$${\left({\kappa }_t^{\begin{array}{|c|}\hline t\\ \hline\end{array}{\displaystyle \frac{1}{\alpha }}}\right)}^{⊞\alpha }\stackrel{\alpha \to +\infty }{\to }{\mathbb{B}}_{1-t}\left({\kappa }_t\right)=sc, indistribution.$$

In fact, for x close enough to $m_0^{{\left({\kappa }_t^{\begin{array}{|c|}\hline t\\ \hline\end{array}{\displaystyle \frac{1}{\alpha }}}\right)}^{⊞\alpha }}=m_0^{{\kappa }_t}=0$, we have

$${\mathcal{V}}_{{\left({\kappa }_t^{\begin{array}{|c|}\hline t\\ \hline\end{array}{\displaystyle \frac{1}{\alpha }}}\right)}^{⊞\alpha }}\left(x\right)\stackrel{\alpha \to +\infty }{\to }{\mathcal{V}}_{{\kappa }_t}\left(x\right)+(1-t)x(x-m_0^{{\kappa }_t})=1={\mathcal{V}}_{sc}\left(x\right).$$

6. Conclusions

A family of probability measure transformations known as t-transformations was presented in [5]. These transformations serve as a basis for creating matching convolution transformations. The links between a measure’s moments and cumulants and those of its t-transformation were investigated, yielding information about the structural changes caused by these transformations. The central limit theorems (CLTs) coming from applying t-transformations to classical and free convolutions were examined, illustrating how these transformations impact the limiting behavior of convoluted measures. In [11], the behavior of VFs within the context of Boolean additive convolution (or t-transformation of measures) was investigated. The link between VFs and the Boolean Bercovici–Pata bijection is also discussed. Understanding this link helps to translate characteristics across different probability frameworks. Furthermore, the study’s discovery of a link between Boolean cumulants and VFs is crucial. The work indicates that Boolean cumulants of specific probability measures are connected to Catalan numbers and Fuss–Catalan numbers, indicating that Boolean convolution is based on a deep combinatorial structure.

In this article, we have continued the investigation of the t-transformation of measures and convolutions. We have provided substantial clarification on the stability of the $\mathcal{FMF}$ of probability measures (respectively, the free counterpart of Letac Mora class) under the t-transformation. Furthermore, novel limit theorems involving both free and Boolean additive convolutions are demonstrated for the $\begin{array}{|c|}\hline t\\ \hline\end{array}$-convolution based on the VF. In fact, the $\mathcal{FMF}$ of measures such as well-known distributions like the semicircular, free Poisson (Marchenko—Pastur), and free binomial laws occupies a central position in free probability theory because of its rich algebraic and combinatorial structure. These measures are characterized by having quadratic VFs and linear free cumulant sequences, making them particularly tractable in both analytic and combinatorial approaches. When t-transformations are applied, they modify the measures’ moment or cumulant sequences systematically, essentially “rescaling” or “shifting” aspects of the distribution without disrupting its deep structural properties. The stability of the $\mathcal{FMF}$ under t-transformations means that, after transformation, the measure remains within the $\mathcal{FMF}$, perhaps with different parameters, but still of the same “type”. Also, the quadratic form of the VF is preserved, though coefficients may change. In addition, the free cumulants remain linear functions of degree up to 2 (i.e., at most quadratic relationships are introduced), which is crucial for maintaining solvable behavior in free convolution semigroups. From an operational standpoint, stability implies robustness: processes modeled by Meixner-type measures (like certain random matrices, large systems in quantum information, or Lévy processes in free probability) will retain their analytical tractability under the action of t-transformations. These findings may be applied in random matrix theory. In fact, the stability suggests that certain ensembles evolve predictably under perturbations related to t-transformations. Moreover, in free stochastic processes, such as free Lévy processes, the fact that distributions remain in the Meixner class ensures that moment-based methods and cumulant techniques remain valid after time evolution or transformations. Furthermore, in non-commutative statistics, it gives a framework to build flexible but analytically controllable families of models that can adapt while preserving key structural properties. In short, the stability of the $\mathcal{FMF}$ under t-transformations shows a powerful balance between flexibility and structural preservation, a key feature for extending classical probabilistic intuitions into the free probability world.

Limit theorems for t-transformations characterize the asymptotic behavior of sequences of measures that have undergone t-transformations, which are frequently combined with convolution procedures. In reality, t-transformations may perform interpolation between classical, free, and Boolean convolutions. As a result, limit theorems using t-transformations aid in describing a continuum of asymptotic behaviors across distinct probabilistic “worlds”. Furthermore, using t-transformations, one may establish limit theorems in which the limiting measure is not necessarily Gaussian (as in the standard CLTs), but rather a semicircular law, a free Meixner law, or another extended distribution. This gives adaptable models for huge systems with non-classical relationships. Furthermore, t-transformations provide a method for controlling higher-order cumulants methodically. Thus, limit theorems following t-transformations enable not only anticipating the form of the limit, but also modifying the “shape” (e.g., variance profile, tail behavior) of the convergence goal. It is commonly understood that classical CLTs illustrate the universality of Gaussian behavior. The free and Boolean limit theorems demonstrate further universality classes. t-transformations produce finer classifications, depending on how the transformation parameter t affects convergence. In random matrix theory, t-transformed limits can simulate transitions between distinct spectral distributions when perturbations vary. Furthermore, in quantum information and operator algebras, limit theorems involving t-transformations provide tools for analyzing fluctuations in vast non-commuting systems. In combinatorics, they provide asymptotic conclusions for random structures (such as trees and graphs) that are naturally expressed using non-commutative cumulants.

In summary, the study of t-transformations of measures and VFs is critical for understanding generalized convolutions in non-commutative probability frameworks such as classical, free, and Boolean probability. A t-transformation manipulates a probability measure in a controlled manner, changing the behavior of its cumulants and moments. VFs, on the other hand, provide critical information on the spread and structure of probability distributions, especially those formed by convolution semigroups. Clarifying the theoretical linkages between t-transformations and VFs allows for more in-depth insights into stochastic processes, random matrix theory, and statistical physics, all of which require understanding the development and interaction of distributions under non-classical conceptions of independence. These connections also open new pathways for constructing models with specific variance profiles and for extending classical limit theorems into non-commutative contexts.