${\mathbb{B}}_{\mathit{t}}$-Transformation and Variance Function

Abstract

This study investigates the ${\mathbb{B}}_t$-transformation of probability measures within the framework of free probability. A primary focus is the invariance under this transformation of two fundamental families: the free Meixner family and the free analog of the Letac–Mora class. In addition, we introduce novel characteristics associated with the ${\mathbb{B}}_t$-transformation, offering refined analytical tools to probe its structural and functional properties. These tools allow us to uncover new and significant properties of several distributions in free probability, including the semicircle, the Marchenko–Pastur, and the free Gamma laws, yielding explicit invariance results and stability conditions. Our findings extend the theoretical understanding of the ${\mathbb{B}}_t$-transformation and provide practical methods for analyzing the dynamics and stability of classical free distributions under this operator.

1. Introduction

A primary aim of probability theory is to characterize how the distributions of random variables evolve under various operations. This work contributes to that goal by investigating transformations of measures, a foundational element indispensable to this line of questioning. Although this issue has been explored in several works [1,2,3,4,5], we focus specifically on the ${\mathbb{B}}_t$-transformation introduced in [6], also called the Belinschi–Nica semigroup. This transformation represents a family that interpolates between the free and Boolean convolutions, and it is the central object of our study. Denote by $\mathbf{P}$ the set of real probability measures. The Cauchy transform ${\mathbf{G}}_{\tau }(·)$ of $\tau \in \mathbf{P}$ is

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

The additive free convolution, denoted as $\varrho ⊞\tau$ for $\tau ,\varrho \in \mathbf{P}$, is characterized by the identity of their free cumulant transforms ${\mathcal{C}}_{\tau }$ [7,8,9]:

$${\mathcal{C}}_{\varrho ⊞\tau }\left(z\right)={\mathcal{C}}_{\varrho }\left(z\right)+{\mathcal{C}}_{\tau }\left(z\right),$$

where ${\mathcal{C}}_{\tau }$ is defined implicitly for w near infinity by the relation

$${\mathcal{C}}_{\tau }\left({\mathbf{G}}_{\tau }\left(w\right)\right)=w-1/{\mathbf{G}}_{\tau }\left(w\right).$$

A measure $\tau \in \mathbf{P}$ is ⊞-infinitely divisible if for each $k\in \mathbb{N}$, ${\tau }_k\in \mathbf{P}$ such that $\tau ={\tau }_k^{⊞k}$, where ${\tau }^{⊞p}$ denotes the p-fold free convolution power. This power is well defined ∀ $p\ge 1$ [10] and satisfies

$${\mathcal{C}}_{{\tau }^{⊞p}}\left(z\right)=p{\mathcal{C}}_{\tau }\left(z\right).$$

In fact, a measure is ⊞-infinitely divisible if and only if ${\tau }^{⊞p}$ is well defined for all $p>0$.

The additive Boolean convolution, denoted as $\rho \uplus \tau$, is defined via the Boolean cumulant transform ${\mathbf{K}}_{\tau }\left(w\right)=w-1/{\mathbf{G}}_{\tau }\left(w\right),$ according to the relation

$${\mathbf{K}}_{\rho \uplus \tau }\left(w\right)={\mathbf{K}}_{\rho }\left(w\right)+{\mathbf{K}}_{\tau }\left(w\right),w\in {\mathbb{C}}^{+}.$$

In contrast to the free case, every probability measure $\tau \in \mathbf{P}$ is ⊎-infinitely divisible ([11], Theorem 3.6); that is, for each $k\in \mathbb{N}$, there is a measure ${\tau }_k$ such that $\tau ={\tau }_k^{\uplus k}$.

For $t\ge 0$, the ${\mathbb{B}}_t$-transformation is the map

$$\begin{array}{ccc}\hfill {\mathbb{B}}_t:\mathbf{P}& \to & \mathbf{P};\tau \mapsto {\mathbb{B}}_t\left(\tau \right)={\left({\tau }^{⊞(1+t)}\right)}^{\uplus {\displaystyle \frac{1}{1+t}}}.\hfill \end{array}$$

Each ${\mathbb{B}}_t$ is shown to be a homomorphism with respect to free multiplicative convolution, thereby revealing deep algebraic compatibility between different convolution structures in free probability. A central result establishes that ${\mathbb{B}}_1$ coincides with the Bercovici–Pata bijection and that for all $t\ge 1$, the images ${\mathbb{B}}_t\left(\tau \right)$ are freely infinitely divisible. The semigroup also admits a probabilistic interpretation connected to free Brownian motion, offering new insights into the interplay between additive, multiplicative, and Boolean frameworks in noncommutative probability. In this context, the authors of [12] realize the Belinschi–Nica semigroup of homomorphisms via free multiplicative subordination and generalize it to broader classes of semigroups, connecting with the complex Burgers’ equation, Boolean and monotone stable laws, and the Markov–Krein transform. On the other hand, Ueda [13] investigates the connections between additive convolution semigroups and max-convolution semigroups, developing formulations that involve the Belinschi–Nica semigroup. The work derives limit theorems for extreme values under classical, free, and Boolean frameworks. Further studies related to ${\mathbb{B}}_t$-transformation in other directions can be found in [14,15,16,17].

On the other hand, in classical probability, the exponential kernel $\exp \left(\theta l\right)$ forms the foundation of natural exponential families (NEFs); see [18,19,20,21]. By analogy, free probability introduces the Cauchy–Stieltjes kernel (CSK) families, which are constructed by the kernel $1/(1-\theta l)$ in place of the exponential. Bryc [22] initiated the investigation of CSK families for compactly supported measures, showing that they can be parameterized by the mean. Within this framework, a family (and its generating measure $\tau$) is uniquely characterized by its variance function (VF) and the mean $m_0^{\tau }$ of $\tau$. The class of quadratic CSK families, those whose VF is a polynomial in the mean of a degree of at most two, was fully described. For this quadratic class, the generating measures were identified as the free Meixner family ($\mathrm{FMF}$) of distributions. This work was extended by Bryc and Hassairi [23] to include measures $\tau$ with support bounded on one side. For this case, they defined the mean domain and a “pseudo-variance” function (PVF), which retains analytical utility akin to that of the VF but lacks a direct probabilistic interpretation. Their work also introduced the concept of reciprocity between two CSK families, established via a relation between the free cumulant transforms of their respective generating measures. This reciprocity was used to characterize a class of cubic CSK families (with one-sided bounded support) which is the reciprocal of the quadratic class. The probability measures generating this cubic class constitute the free analog of the Letac–Mora class.

The study of the ${\mathbb{B}}_t$-transformation of measures gains additional significance when examined through the lens of CSK families of probability measures. Since CSK families provide a noncommutative analogue of NEFs, their structural properties, such as VFs and stability under convolution-type operations, are central in understanding the interplay between the analytic and probabilistic aspects of free probability. The ${\mathbb{B}}_t$-transformation serves as a powerful tool for exploring deformation and stability phenomena within these families. Investigating its action on CSK families not only deepens the comprehension of free harmonic analysis but also sheds light on the robustness of probabilistic structures under measure transformations, offering potential applications to random matrix models and noncommutative statistics.

Moreover, the study of the ${\mathbb{B}}_t$-transformation, in the context of CSK families, has important real-life implications, as it deepens our understanding of how probabilistic models behave under structural perturbations and transformations that preserve essential features such as variance or mean behavior. In practical terms, linking the ${\mathbb{B}}_t$-transformation to applications where one models evolving distributions while maintaining key functional relationships has relevance in areas like quantitative finance, where understanding the stability of variance structures under market shocks is crucial; in data science, where transformations of non-Gaussian data require robust probabilistic frameworks; and in signal and information processing, where noise and uncertainty evolve through nonlinear transformations. By identifying invariant and stable distributions under the ${\mathbb{B}}_t$-transformation, this study offers a deeper mathematical basis for constructing models that remain consistent and reliable when subjected to complex interactions, mirroring the stability requirements of real-world dynamic systems.

This work examines the ${\mathbb{B}}_t$-transformation through the lens of CSK families and their VFs. Section 2 reviews the essential background on VFs. In Section 3 and Section 4, we establish the invariance of two key classes under the ${\mathbb{B}}_t$-transformation: the free Meixner family (FMF) and the free analog of the Letac–Mora class. Section 5 introduces new characteristics related to the ${\mathbb{B}}_t$-transformation and derived from the VF, leading to novel results for fundamental free probability measures, including the semicircle (SC), Marchenko–Pastur (MP), and free Gamma (FG) distributions.

By presenting explicit scalar-level formulas and invariance results, this work complements and advances beyond the more abstract, operator-valued approaches such as those developed in [24]. The results obtained provide practical analytical tools that can be effectively applied in areas like free harmonic analysis and random matrix theory. In essence, the study’s originality lies in introducing a new analytical perspective, based on the CSK framework, for investigating the ${\mathbb{B}}_t$-transformation. Its relevance stems from the way it clarifies and extends the theoretical foundations of free probability, bridging the gap between abstract operator-valued formulations and explicit distributional properties within the scalar setting.

2. The VF Concept

Let ${\mathbf{P}}_{ba}$ and ${\mathbf{P}}_c$ denote the subsets of $\mathbf{P}$ consisting of non-degenerate probability measures with support bounded from above and with compact support, respectively.

For $\tau \in {\mathbf{P}}_{ba}$, the integral

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

is finite for all $\theta \in [0,{\theta }_{+}^{\tau })$, where the upper limit is defined by $1/{\theta }_{+}^{\tau }=\max \{0,sup\mathrm{supp}\left(\tau \right)\}$.

The mean function $K_{\tau }\left(\theta \right)=\int l{\mathcal{P}}_{\theta }^{\tau }\left(dl\right)$ is a bijection from $(0,{\theta }_{+}^{\tau })$ onto an interval $(m_0^{\tau },m_{+}^{\tau })$, known as the (one-sided) mean domain of ${\mathcal{K}}_{+}\left(\tau \right)$; [23]. Denote its inverse by ${\psi }_{\tau }(·)$. For any m in the mean domain, we define ${\mathcal{Q}}_m^{\tau }\left(dl\right)={\mathcal{P}}_{{\psi }_{\tau }\left(m\right)}^{\tau }\left(dl\right)$, which yields the mean-parameterized form of the CSK family:

$${\mathcal{K}}_{+}\left(\tau \right)=\{{\mathcal{Q}}_m^{\tau }\left(dl\right):m\in (m_0^{\tau },m_{+}^{\tau })\}.$$

The boundaries of the mean domain are given by

$$m_0^{\tau }=\underset{\theta \to 0^{+}}{\lim }{K}_{\tau }\left(\theta \right)\mathrm{and}{m}_{+}^{\tau }=B_{\tau }-\underset{w\to B_{\tau }^{+}}{\lim }1/{\mathbf{G}}_{\tau }\left(w\right),$$

where $B_{\tau }=1/{\theta }_{+}^{\tau }$ [23].

When the support of $\tau$ is bounded from below, the corresponding CSK family is denoted by ${\mathcal{K}}_{-}\left(\tau \right)$. In this case, the parameter $\theta$ lies in the interval $({\theta }_{-}^{\tau },0)$, where ${\theta }_{-}^{\tau }$ is defined as $1/A_{\tau }$ or $-\infty$, with $A_{\tau }=\min \{0,inf\mathrm{supp}\left(\tau \right)\}$. The mean domain for ${\mathcal{K}}_{-}\left(\tau \right)$ is the interval $(m_{-}^{\tau },m_0^{\tau })$, where the lower bound is given by $m_{-}^{\tau }=A_{\tau }-1/{\mathbf{G}}_{\tau }\left(A_{\tau }\right)$. If $\tau \in {\mathbf{P}}_c$ (i.e., has compact support), then $\theta$ varies over the full interval $({\theta }_{-}^{\tau },{\theta }_{+}^{\tau })$. The complete, two-sided CSK family is given by the union $\mathcal{K}\left(\tau \right)={\mathcal{K}}_{-}\left(\tau \right)\cup \left\{\tau \right\}\cup {\mathcal{K}}_{+}\left(\tau \right)$.

Let $\tau \in {\mathbf{P}}_{ba}$. The VF defined as

$$m\mapsto {\mathcal{V}}_{\tau }\left(m\right)=\int {(s-m)}^2{\mathcal{Q}}_m^{\tau }\left(ds\right)$$

is a fundamental concept in CSK families, [22]. If $\tau$ does not possess a first moment, then all measures within ${\mathcal{K}}_{+}\left(\tau \right)$ exhibit infinite variance. To address such cases, the notion of a pseudo-variance function (PVF), denoted as ${\mathbb{V}}_{\tau }(·)$, was introduced in [23] and is defined as ${\mathbb{V}}_{\tau }\left(m\right)=m\left(1/{\psi }_{\tau }\left(m\right)-m\right).$ When the mean $m_0^{\tau }=\int l\tau \left(dl\right)$ is finite, the VF ${\mathcal{V}}_{\tau }(·)$ exists, and the two concepts are related by the identity [23]

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

The following remark summarizes relevant details that will be used to prove the main findings of this work.

Remark 1.

(i) 

A probability measure $\tau \in {\mathbf{P}}_{ba}$ is characterized by ${\mathbb{V}}_{\mu }(·)$: With $\varpi =\varpi \left(m\right)=m+{\displaystyle \frac{{\mathbb{V}}_{\tau }\left(m\right)}{m}},$ one has

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

(1)

If $m_0^{\tau }$ is finite, then

$${\mathbf{G}}_{\tau }\left(\varpi \right)={\displaystyle \frac{m-m_0^{\tau }}{{\mathcal{V}}_{\tau }\left(m\right)}}.$$

Thus, ${\mathcal{V}}_{\tau }(·)$ and $m_0^{\tau }$ determine τ.

(ii) 

Let $g:s\mapsto \iota s+\delta$ with $\iota \ne 0$ and $\delta \in \mathbb{R}$. For m near $m_0^{g\left(\tau \right)}=g\left(m_0^{\tau }\right)=\iota m_0^{\tau }+\delta$,

$${\mathbb{V}}_{g\left(\tau \right)}\left(m\right)={\displaystyle \frac{{\iota }^{2}m}{m-\delta }}{\mathbb{V}}_{\tau }\left({\displaystyle \frac{m-\delta }{\iota }}\right).$$

If ${\mathcal{V}}_{\tau }(·)$ exists, then

$${\mathcal{V}}_{g\left(\tau \right)}\left(m\right)={\iota }^2{\mathcal{V}}_{\tau }\left({\displaystyle \frac{m-\delta }{\iota }}\right).$$

(2)

(iii) 

Based on the results from [25], for values of m sufficiently near $m_0^{{\mathbb{B}}_t\left(\tau \right)}=m_0^{\tau }$, the following holds:

$${\mathbb{V}}_{{\mathbb{B}}_t\left(\tau \right)}\left(m\right)={\mathbb{V}}_{\tau }\left(m\right)+t{m}^2.$$

(3)

When $m_0^{\tau }$ is finite,

$${\mathcal{V}}_{{\mathbb{B}}_t\left(\tau \right)}\left(m\right)={\mathcal{V}}_{\tau }\left(m\right)+tm(m-m_0^{\tau }).$$

(4)

The variance in ${\mathbb{B}}_t\left(\tau \right)$ is $\mathrm{Var}\left({\mathbb{B}}_t\left(\tau \right)\right)=\mathrm{Var}\left(\tau \right).$

3. Invariance of the $\mathrm{FMF}$ Under ${\mathbb{B}}_t$-Transformation

Determining the stability of the $\mathrm{FMF}$ under transformations of measures is of fundamental importance, as it sheds light on the robustness of this class with respect to key operations in free probability, such as free convolution powers, t-transformations, and ${\mathbb{B}}_t$-transformations. The $\mathrm{FMF}$ occupies a central role due to its rich algebraic and analytic structure, encompassing many well-known distributions as special cases. Understanding its stability properties not only provides deeper insights into the interplay between free cumulants, VFs, and convolution semigroups but also enhances its applicability in modeling, classification, and asymptotic analysis within free probability and random matrix theory.

The class of quadratic CSK families with

$${\mathcal{V}}_{\tau }\left(m\right)=1+\eta m+\upsilon m^2,\eta \in \mathbb{R},\upsilon \ge -1,m_0^{\tau }=0$$

(5)

was fully characterized in [22]. The relative measures belong to the $\mathrm{FMF}$:

$$\tau \left(d\zeta \right)={\displaystyle \frac{\sqrt{4(1+\upsilon )-{(\zeta -\eta )}^2}}{2\pi (\upsilon {\zeta }^2+\eta \zeta +1)}}{1}_{(\eta -2\sqrt{1+\upsilon },\eta +2\sqrt{1+\upsilon })}\left(\zeta \right)d\zeta +p_1{\delta }_{{\zeta }_1}+p_2{\delta }_{{\zeta }_2}.$$

We have the following:

(i)

If $\upsilon =0,{\eta }^2>1$, then $p_1=1-1/{\eta }^2,x_1=-1/\eta ,p_2=0$.

(ii)

If $\upsilon >0$ and ${\eta }^2>4\upsilon$, then $p_1=\max \{0,1-{\displaystyle \frac{\left|\eta \right|-\sqrt{{\eta }^2-4\upsilon }}{2\upsilon \sqrt{{\eta }^2-4\upsilon }}}\},p_2=0$, and ${\zeta }_1=\pm {\displaystyle \frac{\left|\eta \right|-\sqrt{{\eta }^2-4\upsilon }}{2\upsilon }}$ with the sign opposite to the sign of $\eta$.

(iii)

If $-1\le \upsilon <0$, then

$${\zeta }_{1,2}={\displaystyle \frac{-\eta \pm \sqrt{{\eta }^2-4\upsilon }}{2\upsilon }},p_{1,2}=1+{\displaystyle \frac{\sqrt{{\eta }^2-4\upsilon }\mp \eta }{2\upsilon \sqrt{{\eta }^2-4\upsilon }}}.$$

This result applies to several important classes of measures. Specifically, up to a dilation and a free convolution, the measure $\tau$ is

(i)

The semicircle (SC) measure if $\eta =\upsilon =0$;

(ii)

The Marchenko–Pastur (MP) measure if $\upsilon =0$ and $\eta \ne 0$;

(iii)

The free Pascal (FP) measure if $\upsilon >0$ and ${\eta }^2>4\upsilon$;

(iv)

The free Gamma (FG) measure if $\upsilon >0$ and ${\eta }^2=4\upsilon$;

(v)

The free analog of the hyperbolic (FH) measure if $\upsilon >0$ and ${\eta }^2<4\upsilon$;

(vi)

The free binomial (FB) measure if $-1\le \upsilon <0$.

Next, we show that the FMF is closed under the ${\mathbb{B}}_t$-transformation, meaning that when any member of the FMF is acted upon by ${\mathbb{B}}_t$, the resulting distribution remains within the same family.

Theorem 1.

If $\tau \in \mathrm{FMF}$, then for every $t\ge 0$, ${\mathbb{B}}_t\left(\tau \right)\in \mathrm{FMF}$.

Proof. 

Assume that $\tau \in \mathrm{FMF}$. For $\mathcal{K}\left(\tau \right)$, the VF is

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

(6)

Combining (6) and (4), for m close to $m_0^{{\mathbb{B}}_t\left(\tau \right)}=0$, we obtain

$${\mathcal{V}}_{{\mathbb{B}}_t\left(\tau \right)}\left(m\right)=1+{\eta }^{\prime }m+({\upsilon }^{\prime }+t)m^2,$$

(7)

which corresponds to a VF of the form specified in Equation (5). Then, ${\mathbb{B}}_t\left(\tau \right)\in \mathrm{FMF}$. □

This property of closure implies that the FMF is structurally stable under this transformation, preserving key characteristics such as the VF and cumulant relationships. In practical terms, this means that operations that involve interpolating between Boolean and free convolutions, as encoded by ${\mathbb{B}}_t$, do not generate distributions outside the FMF, making it a robust and self-contained class within free probability. This invariance also allows researchers to predict the behavior of FMF distributions under ${\mathbb{B}}_t$ and provides a foundation for deriving further analytical results, such as explicit formulas for transformed moments, densities, or free cumulants. Overall, the closure property highlights the special structural role of the FMF in the context of transformations that mix Boolean and free probabilistic structures.

In what follows, we illustrate the importance of Theorem 1 by applying it to several fundamental measures.

Corollary 1.

Let $\tau \left(d\xi \right)={\displaystyle \frac{1}{2}}{\delta }_{-1}+{\displaystyle \frac{1}{2}}{\delta }_1$ be the symmetric Bernoulli measure. Then, ${\mathbb{B}}_t\left(\tau \right)$ is

(i) 

An SC measure if $t=1$;

(ii) 

An FH measure with $\upsilon =t-1$ and $\eta =0$ if $t\in (1,+\infty )$;

(iii) 

An FB measure with $-1\le \upsilon =t-1<0$ and $\eta =0$ if $t\in [0,1)$.

Proof. 

We have ${\mathcal{V}}_{\tau }\left(m\right)=1-m^2$. Using (7), we obtain

$${\mathcal{V}}_{{\mathbb{B}}_t\left(\tau \right)}\left(m\right)=1+(t-1)m^2.$$

(8)

Identifying (8) and (5) gives

(i)

If $t=1$, then ${\mathbb{B}}_t\left(\tau \right)$ is an SC measure;

(ii)

If $t>1$, then ${\mathbb{B}}_t\left(\nu \right)$ is an FH measure where $\eta =0$ and $\upsilon =t-1>0$;

(iii)

If $0\le t<1$, then ${\mathbb{B}}_t\left(\nu \right)$ is an FB measure where $\eta =0$ and $-1\le \upsilon =t-1<0$.

□

Corollary 2.

Consider the SC measure

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

(9)

Then, ${\mathbb{B}}_t\left(\tau \right)$ is

(i) 

An SC measure if $t=0$;

(ii) 

An FH measure where $\eta =0$ and $\upsilon =t$ if $t>0$.

Proof. 

We have ${\mathcal{V}}_{\tau }\left(m\right)=1$. Using (7), we obtain

$${\mathcal{V}}_{{\mathbb{B}}_t\left(\tau \right)}\left(m\right)=1+t{m}^2.$$

(10)

Identifying (10) and (5) gives

(i)

If $t=0$, then ${\mathbb{B}}_t\left(\tau \right)=\tau$;

(ii)

If $t>0$, then ${\mathbb{B}}_t\left(\tau \right)$ is an FH measure where $\eta =0$ and $\upsilon =t>0$.

□

Corollary 3. 

For ${\eta }^{\prime }\ne 0$ and ${\upsilon }^{\prime }=0$, the MP measure is

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

(11)

Then, ${\mathbb{B}}_t\left(\tau \right)$ is

(i) 

An MP measure with $\eta ={\eta }^{\prime }$ and $\upsilon ={\upsilon }^{\prime }=0$ if $t=0$;

(ii) 

An FP measure with $\eta ={\eta }^{\prime }$ and $\upsilon =t>0$ if ${\eta }^{\prime 2}>4t$;

(iii) 

An FG measure with $\eta ={\eta }^{\prime }$ and $\upsilon =t>0$ if ${\eta }^{\prime 2}=4t$;

(iv) 

An FH measure with $\eta ={\eta }^{\prime }$ and $\upsilon =t>0$ if ${\eta }^{\prime 2}<4t$.

Proof. 

We have ${\mathcal{V}}_{\tau }\left(m\right)=1+{\eta }^{\prime }m$. Using (7), we obtain

$${\mathcal{V}}_{{\mathbb{B}}_t\left(\tau \right)}\left(m\right)=1+{\eta }^{\prime }m+t{m}^2.$$

(12)

Identifying (12) and (5) gives

(i)

If $t=0$, then ${\mathbb{B}}_t\left(\tau \right)=\tau$;

(ii)

If $t>0$ and ${\eta }^{\prime 2}>4t$, then ${\mathbb{B}}_t\left(\tau \right)$ is an FP measure with $\eta ={\eta }^{\prime }$ and $\upsilon =t>0$;

(iii)

If $t>0$ and ${\eta }^{\prime 2}=4t$, then ${\mathbb{B}}_t\left(\tau \right)$ is an FG measure with $\eta ={\eta }^{\prime }$ and $\upsilon =t>0$;

(iv)

If $t>0$ and ${\eta }^{\prime 2}<4t$, then ${\mathbb{B}}_t\left(\tau \right)$ is an FH measure with $\eta ={\eta }^{\prime }$ and $\upsilon =t>0$.

□

Corollary 4.

For ${\eta }^{\prime }\ne 0$ and ${\upsilon }^{\prime }={\eta }^{\prime 2}/4$, the FG measure is

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

(13)

Then, ${\mathbb{B}}_t\left(\tau \right)$ is

(i) 

An FG measure with $\eta ={\eta }^{\prime }$ and $\upsilon ={\upsilon }^{\prime }$ if $t=0$;

(ii) 

An FH measure with $\eta ={\eta }^{\prime }$ and $\upsilon ={\eta }^{\prime 2}/4+t$ if $t>0$.

Proof. 

We have ${\mathcal{V}}_{\tau }\left(m\right)=1+{\eta }^{\prime }m+{\eta }^{\prime 2}/4m^2$. Using (7), we obtain

$${\mathcal{V}}_{{\mathbb{B}}_t\left(\tau \right)}\left(m\right)=1+{\eta }^{\prime }m+({\eta }^{\prime 2}/4+t)m^2.$$

(14)

Identifying (14) and (5) gives

(i)

If $t=0$, then ${\mathbb{B}}_t\left(\tau \right)=\tau$;

(ii)

If $t>0$, then ${\mathbb{B}}_t\left(\tau \right)$ is an FH measure with $\eta ={\eta }^{\prime }$ and $\upsilon ={\eta }^{\prime 2}/4+t$.

□

4. Invariance of Cubic CSK Families Under ${\mathbb{B}}_t$-Transformation

A class of cubic CSK families with

$${\mathbb{V}}_{\tau }\left(m\right)=m(a{m}^2+bm+c),a>0,m_0^{\tau }=-\infty$$

(15)

was described in [23]. The corresponding measures are

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

(16)

where $P(a,b,c)=1-1/\sqrt{b^2-4ca}$ if $b^2>1+4ca$ and is 0 otherwise.

The most prominent example within this class is the inverse semicircle (ISC) measure

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

(17)

We have ${\mathbb{V}}_{\tau }\left(m\right)=m^3$ and $(m_0^{\tau },m_{+}^{\tau })=(-\infty ,-1)$. It corresponds to $a=1$, $b=0$, and $c=0$. Likewise, explicit descriptions for the free analogs of the remaining five members of the Letac–Mora class are provided in [23], Section 4.2.

Let CPVF denote the class of probability measures whose corresponding PVF takes the cubic form specified in Equation (15). We have

Theorem 2.

If $\tau \in$ CPVF, then for every $t\ge 0$, ${\mathbb{B}}_t\left(\tau \right)\in$ CPVF.

Proof. 

Assume that $\tau \in$ CPVF. The PVF of ${\mathcal{K}}_{+}\left(\tau \right)$ is

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

(18)

Using (18) and (3), for m close sufficiently to $m_0^{{\mathbb{B}}_t\left(\tau \right)}=-\infty$, we obtain

$${\mathbb{V}}_{{\mathbb{B}}_t\left(\tau \right)}\left(m\right)=m(a^{\prime }{m}^2+(b^{\prime }+t)m+c^{\prime }),$$

(19)

which is a PVF of the form (15). Thus, ${\mathbb{B}}_t\left(\nu \right)\in$ CPVF. □

Corollary 5.

(i) 

If τ is the ISC measure (17) with $m_0^{\tau }=-\infty$, $a^{\prime }=1$, $b^{\prime }=c^{\prime }=0$, then ${\mathbb{B}}_1\left(\tau \right)$ is the free Ressel (or free Kendall) measure with $a=1$, $b=1$, and $c=0$.

(ii) 

If τ is the free Abel measure with $m_0^{\tau }=-\infty$, $a^{\prime }=1$, $b^{\prime }=-1$, and $c^{\prime }=0$, then ${\mathbb{B}}_1\left(\tau \right)$ is the ISC measure (17) with $a=1$, $b=c=0$.

Proof. 

(i) We observe that for the PVF of the ISC CSK family given in (18), the parameters take the values $a^{\prime }=1$ and $b^{\prime }=c^{\prime }=0$. Using relation (19), it follows that ${\mathbb{V}}_{{\mathbb{B}}_1\left(\tau \right)}\left(m\right)=m^2(m+1),$ which is precisely the PVF of the free Ressel CSK family (see [23], p. 590). This completes the proof via the application of identity (1).

(ii) If $\tau$ is the free Abel measure, then according to [23], p. 590, the PVF of the corresponding free Abel CSK family, given by (18), has the parameters $a^{\prime }=1$, $b^{\prime }=-1$, and $c^{\prime }=0$. Applying relation (19), we obtain ${\mathbb{V}}_{{\mathbb{B}}_1\left(\tau \right)}\left(m\right)=m^3,$ which is precisely the PVF of the ISC CSK family. This completes the proof via the application of identity (1). □

Within the framework of CSK families and their associated VFs, Section 3 and Section 4 investigate how certain transformations affect the structure of free probability distributions. In particular, we focus on the ${\mathbb{B}}_t$-transformation, a map that modifies measures in a controlled way. Our results show that when this transformation is applied to distributions belonging to the FMF or to the free analog of the Letac–Mora class, the essential structural features of these distributions remain intact. This means that properties such as the form of the VF and the general shape of the corresponding CSK families are preserved under ${\mathbb{B}}_t$. Consequently, the FMF and the free analog of the Letac–Mora class exhibit a form of invariance or stability, making them particularly robust under this type of transformation. This preservation is not only of theoretical interest but also provides a foundation for further analytical studies, allowing researchers to explore new properties, relationships, and applications of these fundamental free distributions in random matrix theory, free harmonic analysis, and noncommutative probability.

5. Further Properties of ${\mathbb{B}}_t$-Transformation

Studying the properties of the ${\mathbb{B}}_t$-transformation is crucial for understanding the analytic and probabilistic mechanisms that connect Boolean and free probability theories. Since ${\mathbb{B}}_t$ forms a semigroup of transformations interpolating between different notions of independence, determining its structural and stability properties provides valuable insight into how probability measures evolve under noncommutative convolutions. Such investigations not only clarify the role of ${\mathbb{B}}_t$ in the hierarchy of transformations but also open pathways for applications in limit theorems, random matrix theory, and the classification of noncommutative distribution families. In this section, we provide various findings pertaining to the SC, MP, and FG measures using the concept of ${\mathbb{B}}_t$-transformation.

Before stating and showing this section’s main results, we first establish some necessary background regarding the mean domain.

Remark 2.

For a measure $\tau \in {\mathbf{P}}_{ba}$, it is well known that $m_0^{\tau }$ behaves predictably under affine transformations g and under free additive convolution powers. Specifically, $m_0^{g\left(\tau \right)}=g\left(m_0^{\tau }\right)$, and for any $s>0$ such that ${\tau }^{⊞s}$ is defined, we have $m_0^{{\tau }^{⊞s}}=s{m}_0^{\tau }$. However, as discussed in [26], Example 3.9, no general closed-form expression exists for $m_{+}^{\tau }$ under such operations. This limitation motivated the authors of [26] to extend the domain of means in a way that preserves the PVF (or the VF, when it exists). The extended upper bound of the mean domain is defined as

$${\mathit{M}}_{+}^{\tau }=inf\left\{m>m_0^{\tau }:{\displaystyle \frac{{\mathbb{V}}_{\tau }\left(m\right)}{m}}<0\right\}.$$

As shown in [26], Section 3.2, ${\mathit{M}}_{+}^{\tau }$ behaves regularly under free additive convolution powers: for any $s>0$ such that ${\tau }^{⊞s}$ is defined, ${\mathit{M}}_{+}^{{\tau }^{⊞s}}=s{\mathit{M}}_{+}^{\tau }$. Moreover, for $\alpha >0$ and $\beta \in \mathbb{R}$, defining the affine transformations ${\mathbf{H}}_{\alpha }\left(x\right)=\alpha x$ and ${\mathbf{T}}_{\beta }\left(x\right)=x+\beta$, we have

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

Throughout the remainder of this section, the mean domain will be considered in the extended form $(m_0^{\tau },{\mathit{M}}_{+}^{\tau })$.

The primary findings of this part are then stated and shown.

Theorem 3.

Let $\tau \in {\mathbf{P}}_{ba}$ with finite $m_0^{\tau }$. Consider the family of measures

$${\mathbb{B}}_t\left({\mathcal{K}}_{+}\left(\tau \right)\right)=\{{\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)\left(ds\right):{\mathcal{Q}}_m^{\tau }\in {\mathcal{K}}_{+}\left(\tau \right)\}.$$

(i) 

For $t\in (0,1)\cup (1,+\infty )$, if ${\mathbb{B}}_t\left({\mathcal{K}}_{+}\left(\tau \right)\right)={\mathcal{K}}_{+}\left({\mathbf{H}}_t\left(\tau \right)\right)$, then up to affinity, τ is an FG measure;

(ii) 

For $t>0$, if ${\mathbb{B}}_t\left({\mathcal{K}}_{+}\left(\tau \right)\right)={\mathcal{K}}_{+}\left({\mathbf{T}}_t\left(\tau \right)\right)$, then up to scaling, τ is an SC measure;

(iii) 

For $t>1$, if ${\mathbb{B}}_t\left({\mathcal{K}}_{+}\left(\tau \right)\right)={\mathcal{K}}_{+}\left({\tau }^{⊞t}\right)$, then up to affinity, τ is an MP measure.

Proof. 

(i) For $t\in (0,1)\cup (1,+\infty )$, suppose that ${\mathbb{B}}_t\left({\mathcal{K}}_{+}\left(\tau \right)\right)={\mathcal{K}}_{+}\left({\mathbf{H}}_t\left(\tau \right)\right)$. So, ∀$m\in (m_0^{\tau },{\mathbf{M}}_{+}^{\tau })$, there is $r\in (m_0^{{\mathbf{H}}_t\left(\tau \right)},{\mathbf{M}}_{+}^{{\mathbf{H}}_t\left(\tau \right)})=(t{m}_0^{\tau },t{\mathbf{M}}_{+}^{\tau })$ so that

$${\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)={\mathcal{Q}}_r^{{\mathbf{H}}_t\left(\tau \right)}.$$

(20)

By means of the free cumulant transformation, relation (20) is

$${\mathcal{C}}_{{\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)}\left(\varsigma \right)={\mathcal{C}}_{{\mathcal{Q}}_r^{{\mathbf{H}}_t\left(\tau \right)}}\left(\varsigma \right),\varsigma \mathrm{close} \mathrm{to}0.$$

(21)

According to [27], we have

$${\mathcal{C}}_{{\mathcal{Q}}_r^{{\mathbf{H}}_t\left(\tau \right)}}\left(\varsigma \right)=m_0^{{\mathcal{Q}}_r^{{\mathbf{H}}_t\left(\tau \right)}}+\mathrm{Var}\left({\mathcal{Q}}_r^{{\mathbf{H}}_t\left(\tau \right)}\right)\varsigma +\varsigma \epsilon \left(\varsigma \right)=r+{\mathcal{V}}_{{\mathbf{H}}_t\left(\tau \right)}\left(r\right)\varsigma +\varsigma \epsilon \left(\varsigma \right),\epsilon \left(\varsigma \right)\stackrel{\varsigma \to 0}{\to }0.$$

(22)

We also have

$${\mathcal{C}}_{{\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)}\left(\varsigma \right)=m_0^{{\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)}+\mathrm{Var}\left({\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)\right)\varsigma +\varsigma {\epsilon }_1\left(\varsigma \right)=m+{\mathcal{V}}_{\tau }\left(m\right)\varsigma +\varsigma {\epsilon }_1\left(\varsigma \right),{\epsilon }_1\left(\varsigma \right)\stackrel{\varsigma \to 0}{\to }0.$$

(23)

Combining (21)–(23), we obtain

$$m+{\mathcal{V}}_{\tau }\left(m\right)\varsigma +\varsigma {\epsilon }_1\left(\varsigma \right)=r+{\mathcal{V}}_{{\mathbf{H}}_t\left(\tau \right)}\left(r\right)\varsigma +\varsigma \epsilon \left(\varsigma \right).$$

This gives that $r=m$ and so ${\mathcal{V}}_{{\mathbf{H}}_t\left(\tau \right)}\left(m\right)={\mathcal{V}}_{\tau }\left(m\right)$. This combined with (2) gives

$$t^2{\mathcal{V}}_{\tau }(m/t)={\mathcal{V}}_{\tau }\left(m\right)\forall m\in (m_0^{\tau },{\mathbf{M}}_{+}^{\tau })\mathrm{and}\forall t>0.$$

(24)

As $\tau$ is (by assumption) non-degenerate, so ${\mathcal{V}}_{\tau }(·)\ne 0$. Equation (24) gives ${\mathcal{V}}_{\tau }\left(m\right)=\lambda m^2$ with $\lambda >0.$

• If $m_0^{\tau }=0$, there is no VF $\mathcal{V}\left(m\right)=\lambda m^2$, with $\lambda >0$. See [28].
• If $m_0^{\tau }\ne 0$, then $\tau$ is the image by $s\mapsto m_0^{\tau }({\displaystyle \frac{{\eta }^{\prime }}{2}}s+1)$ of the FG measure (13) and $\lambda ={\displaystyle \frac{{\eta }^{\prime 2}}{4}}={\upsilon }^{\prime }$. □

Remark 3.

For the FG measure, we have ${\mathit{M}}_{+}^{\tau }=+\infty .$ Suppose that $m_0^{\tau }>0$ and $t\in (0,1)$. So, ∀ $m\in (m_0^{\tau },+\infty )$, one has $r=m\in (m_0^{\tau },+\infty )\subset (t{m}_0^{\tau },+\infty )$. Thus, relation (20) is well defined. If $m_0^{\tau }>0$ and $t>1$, relation ${\mathbb{B}}_t\left({\mathcal{K}}_{+}\left(\tau \right)\right)={\mathcal{K}}_{+}\left({\mathbf{H}}_t\left(\tau \right)\right)$ may be interpreted as ∀ $r\in (m_0^{{\mathbf{H}}_t\left(\tau \right)},{\mathit{M}}_{+}^{{\mathbf{H}}_t\left(\tau \right)})=(t{m}_0^{\tau },t{\mathit{M}}_{+}^{\tau })=(t{m}_0^{\tau },+\infty )$, and there is $m\in (m_0^{\tau },{\mathit{M}}_{+}^{\tau })=(m_0^{\tau },+\infty )$ so that Equation (20) holds. This gives $r=m$. So, ∀$r\in (t{m}_0^{\tau },+\infty )$, one has $m=r\in (t{m}_0^{\tau },+\infty )\subset (m_0^{\tau },+\infty )$. So, relation (20) is well defined. If $m_0^{\tau }<0$ and $t\in (0,1)\cup (1,+\infty )$, the same arguments show that Equation (20) is well defined.

Next, we demonstrate that in Theorem 3(i), the inverse implication is invalid. Assume that $m_0^{\tau }\ne 0$ and $\tau$ is the image by $s\mapsto m_0^{\tau }({\displaystyle \frac{{\eta }^{\prime }}{2}}s+1)$ of the FG measure (13). We show that

$${\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)\ne {\mathcal{Q}}_m^{{\mathbf{H}}_t\left(\tau \right)}.$$

(25)

We have $m_0^{{\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)}=m=m_0^{{\mathcal{Q}}_m^{{\mathbf{H}}_t\left(\tau \right)}}$. Then, $\omega >0$ exists so that ${\mathbb{V}}_{{\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)}(·)$ and ${\mathbb{V}}_{{\mathcal{Q}}_m^{{\mathbf{H}}_t\left(\tau \right)}}(·)$ are well defined on $(m,m+\omega )$.

We know from [28], Equation (41), that

$${\mathbb{V}}_{{\mathcal{Q}}_m^{\tau }}\left(y\right)={\displaystyle \frac{y^3({\eta }^{\prime 2}{m}^2-(y-m)(m-m_0^{\tau }))}{(y-m)(y(m-m_0^{\tau })+m{m}_0^{\tau })}},y\ne m.$$

(26)

Based on (26) and (3), we obtain

$${\mathbb{V}}_{{\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)}\left(y\right)={\mathbb{V}}_{{\mathcal{Q}}_m^{\tau }}\left(y\right)+t{y}^2={\displaystyle \frac{y^3({\eta }^{\prime 2}{m}^2+(t-1)(y-m)(m-m_0^{\tau }))+t{y}^2(y-m)m{m}_0^{\tau }}{(y-m)(y(m-m_0^{\tau })+m{m}_0^{\tau })}}.$$

(27)

On the other hand, we know from [28], Equation (47), that

$${\mathbb{V}}_{{\mathcal{Q}}_m^{{\mathbf{H}}_t\left(\tau \right)}}\left(y\right)={\displaystyle \frac{y^3({\eta }^{\prime 2}{m}^2-(y-m)(m-t{m}_0^{\tau }))}{(y-m)(y(m-t{m}_0^{\tau })+mt{m}_0^{\tau })}},y\ne m.$$

(28)

It is clear from (27) and (28) that ${\mathbb{V}}_{{\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)}\left(y\right)\ne {\mathbb{V}}_{{\mathcal{Q}}_m^{{\mathbf{H}}_t\left(\tau \right)}}\left(y\right)$, ∀$y\in (m,m+\omega )$. This ends the proof of (25) through the use of (1).

(ii) For $t>0$, suppose that ${\mathbb{B}}_t\left({\mathcal{K}}_{+}\left(\tau \right)\right)={\mathcal{K}}_{+}\left({\mathbf{T}}_t\left(\tau \right)\right)$. Then, ∀$r\in (m_0^{{\mathbf{T}}_t\left(\tau \right)},{\mathbf{M}}_{+}^{{\mathbf{T}}_t\left(\tau \right)})=(m_0^{\tau }+t,{\mathbf{M}}_{+}^{\tau }+t)$, and there is $m\in (m_0^{\tau },{\mathbf{M}}_{+}^{\tau })$ so that

$${\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)={\mathcal{Q}}_r^{{\mathbf{T}}_t\left(\tau \right)}.$$

(29)

That is,

$${\mathcal{C}}_{{\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)}\left(\varsigma \right)={\mathcal{C}}_{{\mathcal{Q}}_r^{{\mathbf{T}}_t\left(\tau \right)}}\left(\varsigma \right),\varsigma \mathrm{close} \mathrm{to}0.$$

Or equivalently,

$$m+{\mathcal{V}}_{\tau }\left(m\right)\varsigma +\varsigma {\epsilon }_1\left(\varsigma \right)=r+{\mathcal{V}}_{{\mathbf{T}}_t\left(\tau \right)}\left(r\right)\varsigma +\varsigma {\epsilon }_2\left(\varsigma \right){\epsilon }_1\left(\varsigma \right)\stackrel{\varsigma \to 0}{\to }0\mathrm{and}{\epsilon }_2\left(\varsigma \right)\stackrel{\varsigma \to 0}{\to }0.$$

This implies that $r=m$ and then ${\mathcal{V}}_{{\mathbf{T}}_t\left(\tau \right)}\left(r\right)={\mathcal{V}}_{\tau }\left(r\right)$. This together with (2) implies that

$${\mathcal{V}}_{\tau }(r-t)={\mathcal{V}}_{\tau }\left(r\right),\forall r\in (m_0^{\tau }+t,{\mathbf{M}}_{+}^{\tau }+t)and\forall t>0.$$

(30)

As $\tau$ is (by assumption) non-degenerate, then ${\mathcal{V}}_{\tau }(·)\ne 0$. Thus, relation (30) gives that ${\mathcal{V}}_{\tau }(·)=\lambda$ with $\lambda >0$. It is clear that $\tau$ is the image by $s\mapsto \sqrt{\lambda }s$ of the SC measure (9).

Remark 4.

In the case of the SC measure, we have ${\mathit{M}}_{+}^{\tau }=+\infty .$ Then, ∀ $r\in (m_0^{\tau }+t,+\infty )$, we have $r=m\in (m_0^{\tau }+t,+\infty )\subset (m_0^{\tau },+\infty )$. Thus, relation (29) is well defined.

Next, we demonstrate that in Theorem 3(ii), the inverse implication is also invalid. Assume that $m_0^{\tau }\ne 0$ and $\tau$ is the image by $s\mapsto \sqrt{\lambda }s$ of the SC measure (9). We show that

$${\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)\ne {\mathcal{Q}}_m^{{\mathbf{T}}_t\left(\tau \right)}.$$

(31)

One has $m_0^{{\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)}=m=m_0^{{\mathcal{Q}}_m^{{\mathbf{T}}_t\left(\tau \right)}}$. Then, $\omega >0$ exists so that ${\mathbb{V}}_{{\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)}(·)$ and ${\mathbb{V}}_{{\mathcal{Q}}_m^{{\mathbf{T}}_t\left(\tau \right)}}(·)$ are well defined on $(m,m+\omega )$.

From [29], Equation (2.12), one has

$${\mathbb{V}}_{{\mathcal{Q}}_m^{\tau }}\left(y\right)=y\left({\displaystyle \frac{\lambda }{y-m}}+m_0^{\tau }-m\right).$$

(32)

Based on (32) and (3), we obtain

$${\mathbb{V}}_{{\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)}\left(y\right)={\mathbb{V}}_{{\mathcal{Q}}_m^{\tau }}\left(y\right)+t{y}^2=y\left({\displaystyle \frac{\lambda }{y-m}}+m_0^{\tau }-m+ty\right).$$

(33)

From [29], Equation (2.18), one has

$${\mathbb{V}}_{{\mathcal{Q}}_m^{{\mathbf{T}}_t\left(\tau \right)}}\left(y\right)=y\left({\displaystyle \frac{\lambda }{y-m}}+m_0^{\tau }+t-m\right),y\ne m.$$

(34)

It is clear from (33) and (34) that ${\mathbb{V}}_{{\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)}\left(y\right)\ne {\mathbb{V}}_{{\mathcal{Q}}_m^{{\mathbf{T}}_t\left(\tau \right)}}\left(y\right)$, ∀$y\in (m,m+\omega )$. This ends the proof of (31) through the use of (1).

(iii) For $t>1$, suppose that ${\mathbb{B}}_t\left({\mathcal{K}}_{+}\left(\tau \right)\right)={\mathcal{K}}_{+}\left({\tau }^{⊞t}\right)$. Then, ∀$m\in (m_0^{\tau },{\mathbf{M}}_{+}^{\tau })$, and there is $r\in (m_0^{{\tau }^{⊞t}},{\mathbf{M}}_{+}^{{\tau }^{⊞t}})=(t{m}_0^{\tau },t{\mathbf{M}}_{+}^{\tau })$ so that

$${\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)={\mathcal{Q}}_r^{{\tau }^{⊞t}}.$$

(35)

In terms of the free cumulant transform, this means that

$${\mathcal{C}}_{{\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)}\left(\varsigma \right)={\mathcal{C}}_{{\mathcal{Q}}_r^{{\tau }^{⊞t}}}\left(\varsigma \right),\varsigma \mathrm{close} \mathrm{to}0.$$

Or equivalently,

$$m+{\mathcal{V}}_{\tau }\left(m\right)\varsigma +\varsigma {\epsilon }_1\left(\varsigma \right)=r+{\mathcal{V}}_{{\tau }^{⊞t}}\left(r\right)\varsigma +\varsigma {\epsilon }_3\left(\varsigma \right){\epsilon }_1\left(\varsigma \right)\stackrel{\varsigma \to 0}{\to }0\mathrm{and}{\epsilon }_3\left(\varsigma \right)\stackrel{\varsigma \to 0}{\to }0.$$

Clearly, $r=m$ and then ${\mathcal{V}}_{{\tau }^{⊞t}}\left(m\right)={\mathcal{V}}_{\tau }\left(m\right)$. This accompanied with the relation ${\mathcal{V}}_{{\tau }^{⊞t}}\left(m\right)=t{\mathcal{V}}_{\tau }(m/t)$ (see [23]) gives

$$t{\mathcal{V}}_{\tau }(m/t)={\mathcal{V}}_{\tau }\left(m\right),\forall m\in (m_0^{\tau },{\mathbf{M}}_{+}^{\tau })\mathrm{and}\forall t>1.$$

(36)

As $\tau$ is (by assumption) non-degenerate, ${\mathcal{V}}_{\tau }(·)\ne 0$. Equation (36) provides ${\mathcal{V}}_{\tau }\left(m\right)=\kappa m$ with $\kappa \ne 0$.

• If $m_0^{\tau }=0$, then $\mathcal{V}\left(m\right)=\kappa m$ where $\kappa \ne 0$ is not a VF; see [30], p. 6.
• If $m_0^{\tau }\ne 0$, then $\tau$ is the image by $s\mapsto m_0^{\tau }(1+{\eta }^{\prime }s)$ of the MP measure (11) and $\kappa ={\eta }^{\prime 2}{m}_0^{\tau }$.

Remark 5.

In the case of the MP measure, we have ${\mathit{M}}_{+}^{\tau }=+\infty .$ Suppose that $m_0^{\tau }<0$. Then, ∀ $m\in (m_0^{\tau },+\infty )$, and we have $r=m\in (m_0^{\tau },+\infty )\subset (t{m}_0^{\tau },+\infty )$. Thus, relation (35) is well defined.

If $m_0^{\tau }>0$, relation ${\mathbb{B}}_t\left({\mathcal{K}}_{+}\left(\tau \right)\right)={\mathcal{K}}_{+}\left({\tau }^{⊞t}\right)$ may be interpreted as ∀ $r\in (m_0^{{\tau }^{⊞t}},{\mathit{M}}_{+}^{{\tau }^{⊞t}})=(t{m}_0^{\tau },t{\mathit{M}}_{+}^{\tau })=(t{m}_0^{\tau },+\infty )$, and there is $m\in (m_0^{\tau },{\mathit{M}}_{+}^{\tau })=(m_0^{\tau },+\infty )$ such that relation (35) holds. This will give $m=r$. So, ∀$r\in (t{m}_0^{\tau },+\infty )$, and we have $m=r\in (t{m}_0^{\tau },+\infty )\subset (m_0^{\tau },+\infty )$, and relation (35) is well defined.

Next, in Theorem 3(iii), we demonstrate that the inverse implication is also invalid. Assume that $m_0^{\tau }\ne 0$ and $\tau$ is the image by $s\mapsto m_0^{\tau }(1+{\eta }^{\prime }s)$ of the MP measure (11). We show that

$${\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)\ne {\mathcal{Q}}_m^{{\tau }^{⊞t}}.$$

(37)

One has $m_0^{{\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)}=m=m_0^{{\mathcal{Q}}_m^{{\tau }^{⊞t}}}$. Then, $\omega >0$ so that ${\mathbb{V}}_{{\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)}(·)$ and ${\mathbb{V}}_{{\mathcal{Q}}_m^{{\tau }^{⊞t}}}(·)$ are well defined on $(m,m+\omega )$. One has from [28], Equation (53),

$${\mathbb{V}}_{{\mathcal{Q}}_m^{\tau }}\left(y\right)=y^2\left({\displaystyle \frac{{\eta }^{\prime 2}{m}_0^{\tau }}{y-m}}+{\displaystyle \frac{m_0^{\tau }}{m}}-1\right),y\ne m.$$

(38)

Based on (38) and (3), we obtain

$${\mathbb{V}}_{{\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)}\left(y\right)={\mathbb{V}}_{{\mathcal{Q}}_m^{\tau }}\left(y\right)+t{y}^2=y^2\left({\displaystyle \frac{{\eta }^{\prime 2}{m}_0^{\tau }}{y-m}}+{\displaystyle \frac{m_0^{\tau }}{m}}+t-1\right).$$

(39)

From [28], Equation (59), one has

$${\mathbb{V}}_{{\mathcal{Q}}_m^{{\tau }^{⊞t}}}\left(y\right)=y^2\left({\displaystyle \frac{{\eta }^{\prime 2}{m}_0^{\tau }}{y-m}}+{\displaystyle \frac{t{m}_0^{\tau }}{m}}-1\right),y\ne m.$$

(40)

It is clear from (39) and (40) that ${\mathbb{V}}_{{\mathbb{B}}_t\left({\mathcal{Q}}_m^{\tau }\right)}\left(y\right)\ne {\mathbb{V}}_{{\mathcal{Q}}_m^{{\tau }^{⊞t}}}\left(y\right)$, ∀$y\in (m,m+\omega )$. This ends the proof of (37) through the use of (1).

Theorem 3 characterizes the distributions that are stable under the action of the ${\mathbb{B}}_t$-transformation when applied to the CSK family of measures generated by a given probability measure $\tau$ with a finite first moment. Specifically, it identifies the conditions under which the transformed family ${\mathbb{B}}_t\left({\mathcal{K}}_{+}\left(\tau \right)\right)$ coincides with another canonical CSK family. Part $\left(i\right)$ shows that if for $t\in (0,1)\cup (1,+\infty )$ the transformed family corresponds to a ${\mathbf{H}}_t$-type transformation, then $\tau$ must be an FG measure, up to an affine transformation. Part $\left(ii\right)$ states that if for $t>0$ the transformed family aligns with a ${\mathbf{T}}_t$-type transformation, then $\tau$ must be an SC measure, up to scaling. Finally, part (iii) asserts that if for $t>1$ the transformed family equals the CSK family generated by t-fold free additive convolution of $\tau$, then $\tau$ must be an MP measure, again up to an affine transformation. Overall, this theorem highlights that the ${\mathbb{B}}_t$-transformation can serve as a rigidity test: only certain fundamental free probability distributions retain their structure under specific ${\mathbb{B}}_t$ operations.

6. Conclusions

While the operator-valued extension of the ${\mathbb{B}}_t$-transformation developed in [24,31] investigates free Brownian motion and free convolution semigroups in a multiplicative operator-valued framework and in operator-valued infinitesimal Boolean and monotone independence, the present study focuses on the scalar setting, providing explicit analytical results for fundamental free distributions such as the SC, MP, and FG laws. Our main contributions include new invariance and stability results for the FMF and the free Letac–Mora class under ${\mathbb{B}}_t$, together with novel analytical tools for examining moments, VFs, and domains of means. Unlike the general operator-valued approach, our scalar analysis yields concrete distributional properties and explicit formulas, offering practical insights and applications that are not directly accessible in the operator-valued framework. Moreover, while the previous work [28] centered on the $T_a$-transformation of measures [32] and its influence on free probability distributions, the current study extends this line of research by exploring the ${\mathbb{B}}_t$-transformation within the context of CSK families, thereby revealing deeper structural and analytical connections in free probability.

As a perspective, it would be highly valuable to investigate the ${\mathbb{B}}_t$-transformation in conjunction with other transformations of measures, such as the $V_a$-transformation [33] or the $(a,b)$-transformation [34]. Combining these operators provides a rich analytical framework for exploring the joint dynamics of free distributions and understanding how VFs, domains of means, and cumulant structures evolve under successive or composite transformations. Such an integrated approach can reveal new invariance properties, stability conditions, and structural interdependencies within CSK families that might remain obscured when each transformation is analyzed in isolation. Moreover, future research could extend the present analysis in several complementary directions. One promising line is to apply the ${\mathbb{B}}_t$-transformation to other noncommutative distribution families beyond the free Meixner and free Letac–Mora classes, thereby uncovering broader invariance phenomena and structural patterns. Another important extension concerns the development of operator-valued analogs of the current framework, which would make the ${\mathbb{B}}_t$-approach applicable to systems involving operator-valued random variables. Finally, establishing explicit connections between the ${\mathbb{B}}_t$-transformation and random matrix ensembles could yield new insights into spectral dynamics and non-classical limit theorems, thus strengthening the link between free probability, asymptotic spectral theory, and applications in high-dimensional data analysis and mathematical physics.