On $\mathbb{T}$-Transformation of Probability Measures

Abstract

The concept of the $(\mathbb{T}=(s,t\left)\right)$-transformation of probability measures, introduced for $s>0$ and $t\in \mathbb{R}$, is examined in this work from the perspective of Cauchy–Stieltjes kernel (CSK) families and their related variance functions (VFs). We calculate the VF formula under the $\mathbb{T}$-transformation of measures. Furthermore, the stability of the free Meixner family ($\mathcal{FMF}$) of probability measures under the $(\mathbb{T}=(s,t\left)\right)$-transformation is significantly shown based on this formula. Additionally, the Wigner’s semicircle CSK family is given a novel characterization based on the $(1,t)$-transformation of probability measures.

1. Introduction

Free probability theory, introduced by Dan Voiculescu in the 1980s, is a mathematical framework that extends classical probability theory to noncommutative contexts, such as operator algebras and random matrix theory. It replaces the classical notion of independence with freedom (or free independence), a concept that describes how noncommutative random variables interact. Free probability theory has revolutionized the study of noncommutative structures by providing a robust probabilistic framework. It has become an essential tool in mathematics and theoretical physics, offering deep insights into both abstract algebraic structures and applied areas such as random matrices and data science.

On the other hand, the transformation of probability measures is a fundamental tool in many fields of mathematics and applied sciences, including probability theory, statistics, and finance. This process allows us to manipulate and change probability distributions to better model complex phenomena or to facilitate the calculation of certain theoretical results. In other words, transforming a probability measure amounts to applying a change function that modifies the way events are perceived or measured in a given probabilistic space, while preserving the essential properties related to probabilities. This topic finds practical application in fields such as stochastic simulations, optimization, and even financial risk modeling.

The topic of transforming probability measures has been explored and expanded within the framework of free probability theory in several ways in numerous papers; see [1,2,3,4,5,6,7]. In this context, a two-parameter transformation $U^{\mathbb{T}}$ of real probability measures is introduced in [8], which is a generalization of the t-transformation defined in [9,10]. For $s>0$ and $t\in \mathbb{R}$, we consider a pair of numbers $\mathbb{T}=(s,t)$ and define the $(\mathbb{T}=(s,t\left)\right)$-transformation by

$${\displaystyle \frac{1}{{\mathcal{G}}_{U^{\mathbb{T}}\left(\nu \right)}\left(z\right)}}={\displaystyle \frac{s}{{\mathcal{G}}_{\nu }\left(z\right)}}+(1-s)z+(t-s)C_{\nu },$$

(1)

where

$${\mathcal{G}}_{\nu }\left(z\right)=\int {\displaystyle \frac{\nu \left(dr\right)}{z-r}},z\in \mathbb{C}\setminus supp\left(\nu \right),$$

represents the Cauchy–Stieltjes transform of measure $\nu$ and $C_{\nu }=\Re \left({\displaystyle \frac{1}{{\mathcal{G}}_{\nu }\left(i\right)}}\right)$ is the so-called Nevanlinna constant of $\nu$.

If $t=s>0$, the transform $U^{\mathbb{T}}(·)$ is nothing but the t-transformation [9,10]. In the case $s=1$, Equation (1) reduces to

$${\displaystyle \frac{1}{{\mathcal{G}}_{U^{(1,t)}\left(\nu \right)}\left(z\right)}}={\displaystyle \frac{1}{{\mathcal{G}}_{\nu }\left(z\right)}}+(t-1)C_{\nu }.$$

(2)

In this paper, we are interested in the study of the $(\mathbb{T}=(s,t\left)\right)$-transformation of probability measures from the perspective of Cauchy–Stieltjes kernel (CSK) families and their corresponding variance functions (VFs). In order to provide the reader with background information, Section 2 will outline some facts about CSK families. The formula for the VF under the $(\mathbb{T}=(s,t\left)\right)$-transformation is demonstrated in Section 3. Based on this expression, the stability of the free Meixner family ($\mathcal{FMF}$) of probability measures under the $(\mathbb{T}=(s,t\left)\right)$-transformation is significantly justified in Section 4: We show that the $\mathbb{T}$-transformation of any member of the $\mathcal{FMF}$ remains in the $\mathcal{FMF}$. In Section 5, a new characterization is provided for the Wigner’s semicircle CSK family-based $(1,t)$-transformation of probability measures.

2. CSK Families

The framework of Cauchy–Stieltjes Kernel (CSK) families, in the context of free probability, has been newly introduced, similarly to natural exponential families, by incorporating the Cauchy–Stieltjes kernel ${\displaystyle \frac{1}{1-\vartheta x}}$, replacing $exp\left(\vartheta x\right)$: the exponential kernel. The CSK families have been examined in [11] for compactly supported measures. Extended characteristics are proved in [12], exploring measures with one-sided support boundary, say, from above. Denote by ${\mathbb{P}}_{ba}$ the set of (non-degenerate) probability measures with one-sided support boundary from above. For $\mu \in {\mathbb{P}}_{ba}$, the function

$${\mathcal{M}}_{\mu }\left(\vartheta \right)=\int {\displaystyle \frac{\mu \left(dx\right)}{1-\vartheta x}}$$

is defined as ∀$\vartheta \in [0,{\vartheta }_{+}^{\mu })$, where ${\displaystyle \frac{1}{{\vartheta }_{+}^{\mu }}}=max\{sup\mathrm{supp}\left(\mu \right),0\}$. The set of probabilities

$${\mathbb{F}}_{+}\left(\mu \right)=\left\{P_{\vartheta }^{\mu }\left(dx\right)={\displaystyle \frac{\mu \left(dx\right)}{{\mathcal{M}}_{\mu }\left(\vartheta \right)(1-\vartheta x)}}:0<\vartheta <{\vartheta }_{+}^{\mu }\right\}$$

is called the one-sided CSK family induced by $\mu$.

The map $\vartheta \mapsto k_{\mu }\left(\vartheta \right)=\int x{P}_{\vartheta }^{\mu }\left(dx\right)$ is called the mean function. It is strictly increasing on the interval $(0,{\vartheta }_{+}^{\mu })$ (see [12], pp. 579–580) and

$$k_{\mu }\left(\vartheta \right)={\displaystyle \frac{{\mathcal{M}}_{\mu }\left(\vartheta \right)-1}{\vartheta {\mathcal{M}}_{\mu }\left(\vartheta \right)}}.$$

(3)

The domain of means for ${\mathbb{F}}_{+}\left(\mu \right)$ is the interval $(m_0^{\mu },m_{+}^{\mu })=k_{\mu }\left((0,{\vartheta }_{+}^{\mu })\right)$. For $m_0^{\mu }<m<m_{+}^{\mu }$, let $Q_m^{\mu }\left(dx\right)=P_{{\psi }_{\mu }\left(m\right)}^{\mu }\left(dx\right)$, where ${\psi }_{\mu }(·)$ represents the inverse function of $k_{\mu }(·)$. The parametrization by the mean of ${\mathbb{F}}_{+}\left(\mu \right)$ is

$${\mathbb{F}}_{+}\left(\mu \right)=\{Q_m^{\mu }\left(dx\right):m\in (m_0^{\mu },m_{+}^{\mu })\}.$$

We know from [12] that $m_0^{\mu }={lim}_{\vartheta \to 0^{+}}{k}_{\mu }\left(\vartheta \right)$ and $m_{+}^{\mu }=\mathbb{B}-{lim}_{z\to {\mathbb{B}}^{+}}{\displaystyle \frac{1}{{\mathcal{G}}_{\mu }\left(z\right)}}$, with $\mathbb{B}=\mathbb{B}\left(\mu \right)={\displaystyle \frac{1}{{\vartheta }_{+}^{\mu }}}.$

If $\mu$ possesses a one-sided support boundary from below, the CSK family is represented as ${\mathbb{F}}_{-}\left(\mu \right)$. We have $\vartheta \in ({\vartheta }_{-}^{\mu },0)$, where ${\vartheta }_{-}^{\mu }$ is equal to $-\infty$ or $1/\mathbb{A}\left(\mu \right)$ with $\mathbb{A}\left(\mu \right)=\mathbb{A}=min\{infsupp(\mu ),0\}$. For ${\mathbb{F}}_{-}\left(\mu \right)$, $(m_{-}^{\mu },m_0^{\mu })$ is the domain of means with $m_{-}^{\mu }=\mathbb{A}-1/{\mathcal{G}}_{\mu }\left(\mathbb{A}\right)$. If the support of $\mu$ is compact, then $\mathbb{F}\left(\mu \right)={\mathbb{F}}_{-}\left(\mu \right)\cup \left\{\mu \right\}\cup {\mathbb{F}}_{+}\left(\mu \right)$ is the two-sided CSK family.

For $\mu \in {\mathbb{P}}_{ba}$, the function

$$m\mapsto {\mathcal{V}}_{\mu }\left(m\right)=\int {(r-m)}^2{Q}_m^{\mu }\left(dr\right)$$

is called the variance function (VF) of ${\mathbb{F}}_{+}\left(\mu \right)$; see [11]. If the moment of order one of $\mu$ does not exist, the variance is infinite for all members of ${\mathbb{F}}_{+}\left(\mu \right)$. The following alternative is presented in [12] as

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

and is called pseudo-variance function (PVF) of ${\mathbb{F}}_{+}\left(\mu \right)$. If $m_0^{\mu }=\int x\mu \left(dx\right)$ exists finitely, then the VF exists, and (see [12])

$${\mathbb{V}}_{\mu }\left(m\right)=m{\mathcal{V}}_{\mu }\left(m\right)/(m-m_0^{\mu }).$$

(4)

Remark 1.

(i) We have $Q_m^{\mu }\left(dr\right)=g_{\mu }(r,m)\mu \left(dr\right)$ with

$$g_{\mu }(r,m):=\left\{\begin{array}{cc}{\mathbb{V}}_{\mu }\left(m\right)/({\mathbb{V}}_{\mu }\left(m\right)+m(m-r)),m\ne 0\hfill & ;\hfill \\ 1,m=0,{\mathbb{V}}_{\mu }\left(0\right)\ne 0\hfill & ;\hfill \\ {\mathbb{V}}_{\mu }^{\prime }\left(0\right)/({\mathbb{V}}_{\mu }^{\prime }\left(0\right)-r),m=0,{\mathbb{V}}_{\mu }\left(0\right)=0\hfill & .\hfill \end{array}\right.$$

(5)

(ii) 

${\mathbb{V}}_{\mu }(·)$ determine μ: Consider $\mathrm{\Lambda }=\mathrm{\Lambda }\left(m\right)=m+{\displaystyle \frac{{\mathbb{V}}_{\mu }\left(m\right)}{m}},$ then

$${\mathcal{G}}_{\mu }\left(\mathrm{\Lambda }\right)={\displaystyle \frac{m}{{\mathbb{V}}_{\mu }\left(m\right)}}.$$

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

$${\mathcal{G}}_{\mu }\left(\mathrm{\Lambda }\right)={\displaystyle \frac{m-m_0^{\mu }}{{\mathcal{V}}_{\mu }\left(m\right)}}.$$

(6)

Thus, ${\mathcal{V}}_{\mu }(·)$ and $m_0^{\mu }$ characterizes μ.

(iii) 

Let ${\tau }_{\alpha ,\eta }:x\mapsto \alpha x+\eta$ with $\alpha \ne 0$ and $\eta \in \mathbb{R}$. Then, $\mathrm{\forall }$m close sufficiently to $m_0^{{\tau }_{\alpha ,\eta }\left(\mu \right)}={\tau }_{\alpha ,\eta }\left(m_0^{\mu }\right)=\alpha m_0^{\mu }+\eta$

$${\mathbb{V}}_{{\tau }_{\alpha ,\eta }\left(\mu \right)}\left(m\right)={\displaystyle \frac{{\alpha }^{2}m}{m-\eta }}{\mathbb{V}}_{\mu }\left({\displaystyle \frac{m-\eta }{\alpha }}\right).$$

(7)

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

$${\mathcal{V}}_{{\tau }_{\alpha ,\eta }\left(\mu \right)}\left(m\right)={\alpha }^2{\mathcal{V}}_{\mu }\left({\displaystyle \frac{m-\eta }{\alpha }}\right).$$

(8)

3. $(\mathit{s},\mathit{t})$-Transformation and VF

This section presents the effect of the $(s,t)$-transformation of $\mu \in {\mathbb{P}}_{ba}$ on the associated CSK family.

Proposition 1.

Let $\mu \in {\mathbb{P}}_{ba}$. Then, $\forall \vartheta >0$ close enough to 0, and we have

$$k_{U^{(s,t)}\left(\mu \right)}\left(\vartheta \right)=s{k}_{\mu }\left(\vartheta \right)+(s-t)C_{\mu },$$

(9)

with $C_{\mu }=\Re \left({\displaystyle \frac{1}{{\mathcal{G}}_{\mu }\left(i\right)}}\right)$ as the so-called Nevanlinna constant of the probability measure μ.

Proof. 

From the fact that ${\mathcal{M}}_{\mu }\left(\vartheta \right)={\displaystyle \frac{1}{\vartheta }}{\mathcal{G}}_{\mu }\left({\displaystyle \frac{1}{\vartheta }}\right)$, we see from (1) that

$${\mathcal{M}}_{U^{(s,t)}\left(\mu \right)}\left(\vartheta \right)={\displaystyle \frac{{\mathcal{M}}_{\mu }\left(\vartheta \right)}{s+((1-s)+(t-s)C_{\mu }\vartheta ){\mathcal{M}}_{\mu }\left(\vartheta \right)}}.$$

(10)

It is clear that ${\mathcal{M}}_{U^{(s,t)}\left(\mu \right)}(·)$ is well defined ∀$\vartheta >0$ close enough to 0. Combining (3) with (10), we obtain

$$k_{U^{(s,t)}\left(\mu \right)}\left(\vartheta \right)={\displaystyle \frac{M_{U^{(s,t)}\left(\mu \right)}\left(\vartheta \right)-1}{\vartheta {\mathcal{M}}_{U^{(s,t)}\left(\mu \right)}\left(\vartheta \right)}}=s{\displaystyle \frac{{\mathcal{M}}_{\mu }\left(\vartheta \right)-1}{\vartheta {\mathcal{M}}_{\mu }\left(\vartheta \right)}}+(s-t)C_{\mu }=s{k}_{\mu }\left(\vartheta \right)+(s-t)C_{\mu }.$$

□

Next, we present how the $(s,t)$-transformation of $\mu$ affects ${\mathcal{V}}_{\mu }(·)$.

Theorem 1.

Let $\mu \in {\mathbb{P}}_{ba}$. Then,

$$m_0^{U^{(s,t)}\left(\mu \right)}=s{m}_0^{\mu }+(s-t)C_{\mu },$$

(11)

and $\forall \tilde{m}>m_0^{U^{(s,t)}\left(\mu \right)}$ close enough to $m_0^{U^{(s,t)}\left(\mu \right)}$, we have

$${\mathbb{V}}_{U^{(s,t)}\left(\mu \right)}\left(\tilde{m}\right)={\displaystyle \frac{s\tilde{m}}{\tilde{m}+(t-s)C_{\mu }}}{\mathbb{V}}_{\mu }\left({\displaystyle \frac{\tilde{m}+(t-s)C_{\mu }}{s}}\right)+{\displaystyle \frac{\tilde{m}}{s}}\left(\tilde{m}(1-s)+(t-s)C_{\mu }\right).$$

(12)

where $C_{\mu }=\Re \left({\displaystyle \frac{1}{{\mathcal{G}}_{\mu }\left(i\right)}}\right)$ is the so-called Nevanlinna constant of the measure μ. Furthermore, if $m_0^{\mu }$ is finite, then

$${\mathcal{V}}_{U^{(s,t)}\left(\mu \right)}\left(\tilde{m}\right)=s{\mathcal{V}}_{\mu }\left({\displaystyle \frac{\tilde{m}+(t-s)C_{\mu }}{s}}\right)+{\displaystyle \frac{\tilde{m}(1-s)+(t-s)C_{\mu }}{s}}(\tilde{m}+(t-s)C_{\mu }-s{m}_0^{\mu }).$$

(13)

Proof. 

∀$\vartheta >0$ close enough to 0, denote by $m=\int x{P}_{\vartheta }^{\mu }\left(dx\right)$ and $\tilde{m}=\int x{P}_{\vartheta }^{U^{(s,t)}\left(\mu \right)}\left(dx\right)$. From (9), one can see that

$$\tilde{m}=sm+(s-t)C_{\mu },$$

(14)

and

$$m_0^{U^{(s,t)}\left(\mu \right)}=\underset{\vartheta ⟶0}{lim}{k}_{U^{(s,t)}\left(\mu \right)}\left(\vartheta \right)=s{m}_0^{\mu }+(s-t)C_{\mu }.$$

∀$\vartheta >0$ close enough to 0, we have

$${\psi }_{U^{(s,t)}\left(\mu \right)}\left(\tilde{m}\right)=\vartheta ={\psi }_{\mu }\left(m\right).$$

This may be written as

$${\displaystyle \frac{{\mathbb{V}}_{U^{(s,t)}\left(\mu \right)}\left(\tilde{m}\right)}{\tilde{m}}}+\tilde{m}={\displaystyle \frac{{\mathbb{V}}_{\mu }\left(m\right)}{m}}+m.$$

(15)

Expressing m as a function of $\tilde{m}$ from (14) and inserting it in (15), we obtain relation (12).

Furthermore, if $m_0^{\mu }$ is finite, then ${\mathcal{V}}_{\mu }(·)$ and ${\mathcal{V}}_{U^{(s,t)}\left(\mu \right)}(·)$ exist, and relation (13) follows easily by combining (12) and (4). □

4. Notes on the $\mathcal{FMF}$

In this section, using the VF, we show that the $\mathcal{FMF}$ is invariant when applying the $(s,t)$-transformation of measures. In fact, the quadratic class of CSK families having VF

$${\mathcal{V}}_{\mu }\left(m\right)=1+am+b{m}^2,a\in \mathbb{R},b\ge -1$$

(16)

with $m_0^{\mu }=0$ are described in [11]. The associated laws are the $\mathcal{FMF}$:

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

(17)

We have the following:

(i)

If $b=0,a^2>1$, then $p_1=1-1/a^2,x_1=-1/a,p_2=0$.

(ii)

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

(iii)

If $-1\le b<0$, then there are two atoms at

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

This finding covers important measures. Up to dilation and convolution, measure $\mu$ is as follows:

(i)

The Wigner’s semicircle (free Gaussian) law if $a=b=0$.

(ii)

The Marchenko–Pastur (free Poisson)-type law if $b=0$ and $a\ne 0$.

(iii)

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

(iv)

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

(v)

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

(vi)

The free binomial type law if $-1\le b<0$.

Next, we show the invariance of the $\mathcal{FMF}$ when applying the $(s,t)$-deformation.

Theorem 2.

If $\nu \in \mathcal{FMF}$, then for $s>0$ and $t\in \mathbb{R}$, ${\tau }_{{\displaystyle \frac{1}{\sqrt{s}}},{\displaystyle \frac{(t-s)C_{\nu }}{\sqrt{s}}}}\left(U^{(s,t)}\left(\nu \right)\right)\in \mathcal{FMF}$ where ${\tau }_{{\displaystyle \frac{1}{\sqrt{s}}},{\displaystyle \frac{(t-s)C_{\nu }}{\sqrt{s}}}}\left(x\right)={\displaystyle \frac{1}{\sqrt{s}}}(x+(t-s)C_{\nu })$.

Proof. 

Assume that $\nu \in \mathcal{FMF}$. Then

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

(18)

Combining (18), (8), and (13), ∀m close to $m_0^{{\tau }_{{\displaystyle \frac{1}{\sqrt{s}}},{\displaystyle \frac{(t-s)C_{\nu }}{\sqrt{s}}}}\left(U^{(s,t)}\left(\nu \right)\right)}=0$, we obtain

$${\mathcal{V}}_{{\tau }_{{\displaystyle \frac{1}{\sqrt{s}}},{\displaystyle \frac{(t-s)C_{\nu }}{\sqrt{s}}}}\left(U^{(s,t)}\left(\nu \right)\right)}\left(m\right)=1+\left({\displaystyle \frac{a^{\prime }+(t-s)C_{\nu }}{\sqrt{s}}}\right)m+\left({\displaystyle \frac{b^{\prime }+1-s}{s}}\right)m^2,$$

(19)

which is a VF of the form (16). Then, ${\tau }_{{\displaystyle \frac{1}{\sqrt{s}}},{\displaystyle \frac{(t-s)C_{\nu }}{\sqrt{s}}}}\left(U^{(s,t)}\left(\nu \right)\right)$ belongs to $\mathcal{FMF}$. □

Next, by using some particular measures, the importance of Theorem 2 is illustrated.

Corollary 1.

Let $\nu \left(d\zeta \right)={\displaystyle \frac{1}{2}}{\delta }_{-1}+{\displaystyle \frac{1}{2}}{\delta }_1$. Then, ${\tau }_{{\displaystyle \frac{1}{\sqrt{s}}},{\displaystyle \frac{(t-s)C_{\nu }}{\sqrt{s}}}}\left(U^{(s,t)}\left(\nu \right)\right)=\nu$.

Proof. 

The VF of $\mathbb{F}\left(\nu \right)$ is given by (18) for $a^{\prime }=0$ and $b^{\prime }=-1$. In addition, from the fact that ${\mathcal{G}}_{\nu }\left(z\right)={\displaystyle \frac{z}{z^2-1}}$, we obtain $C_{\nu }=\Re \left(1/{\mathcal{G}}_{\nu }\left(i\right)\right)=0$. From (19), we have that ${\mathcal{V}}_{{\tau }_{{\displaystyle \frac{1}{\sqrt{s}}},{\displaystyle \frac{(t-s)C_{\nu }}{\sqrt{s}}}}\left(U^{(s,t)}\left(\nu \right)\right)}\left(m\right)=1-m^2={\mathcal{V}}_{\nu }\left(m\right)$. The use of (6) achieves the proof. □

Corollary 2.

Consider the semicircle law

$${\displaystyle W\left(d\zeta \right)=\frac{\sqrt{4-{\zeta }^2}}{2\pi }{\mathbf{1}}_{(-2,2)}\left(\zeta \right)d\zeta .}$$

(20)

Then, ${\tau }_{{\displaystyle \frac{1}{\sqrt{s}}},{\displaystyle \frac{(t-s)C_W}{\sqrt{s}}}}\left(U^{(s,t)}\left(W\right)\right)$ is as follows:

(i) 

The semicircle law if $s=1$.

(ii) 

The free analog of hyperbolic-type law of the form (17) with $a=0$ and $b=1/s-1>0$ if $s\in (0,1)$.

(iii) 

The free binomial-type law of the form (17) with $a=0$ and $-1\le b=1/s-1<0$ if $s\in (1,+\infty )$.

Proof. 

We know that $a^{\prime }=b^{\prime }=0$ for the VF of the semicircle CSK family of the form (18). In addition, from the fact that ${\mathcal{G}}_W\left(z\right)={\displaystyle \frac{z-\sqrt{z^2-4}}{2}}$, we obtain $C_W=\Re \left({\displaystyle \frac{1}{{\mathcal{G}}_W\left(i\right)}}\right)=0$. Relation (19) gives

$${\mathcal{V}}_{{\tau }_{{\displaystyle \frac{1}{\sqrt{s}}},0}\left(U^{(s,t)}\left(W\right)\right)}\left(m\right)=1+\left({\displaystyle \frac{1}{s}}-1\right)m^2.$$

(21)

An identification between (21) and (16) provides the following:

(i)

If $s=1$, then ${\tau }_{{\displaystyle \frac{1}{\sqrt{s}}},0}\left(U^{(s,t)}\left(W\right)\right)=W$.

(ii)

If $s\in (0,1)$, then ${\tau }_{{\displaystyle \frac{1}{\sqrt{s}}},0}\left(U^{(s,t)}\left(W\right)\right)$ is of the free analog of hyperbolic-type law of the form (17) with $a=0$ and $b=1/s-1>0$.

(iii)

If $s\in (1,+\infty )$, then ${\tau }_{{\displaystyle \frac{1}{\sqrt{s}}},0}\left(U^{(s,t)}\left(W\right)\right)$ is of the free binomial-type law of the form (17) with $a=0$ and $-1\le b=1/s-1<0$.

□

5. A Characterization of the Semicircle CSK Family

In this section, we present a new characteristic property of the semicircle law. We investigate the stability of CSK families under the $U^{(1,t)}$-transformation of probability measures. Let ${\mathbb{F}}_{+}\left(\mu \right)=\{Q_m^{\mu }\left(dx\right):m\in (m_0^{\mu },m_{+}^{\mu })\}$ be the CSK family induced by $\mu \in {\mathbb{P}}_{ba}$. We introduce a new family of probability measures

$$U^{(1,t)}\left({\mathbb{F}}_{+}\left(\mu \right)\right)=\{U^{(1,t)}\left(Q_m^{\mu }\right)\left(dx\right):m\in (m_0^{\mu },m_{+}^{\mu })\}.$$

We prove that $U^{(1,t)}\left({\mathbb{F}}_{+}\left(\mu \right)\right)$ is a re-parametrization of ${\mathbb{F}}_{+}\left(\mu \right)$ (i.e., $U^{(1,t)}\left({\mathbb{F}}_{+}\left(\mu \right)\right)={\mathbb{F}}_{+}\left(\mu \right)$) if and only if $\mu$ is (up to affinity) of the semicircle-type measure given by (20).

Now, the main finding of this paragraph is stated and proved.

Theorem 3.

Let $\mu \in {\mathbb{P}}_{ba}$. The following assertions are equivalent:

(i) 

$U^{(1,t)}\left(Q_m^{\mu }\right)=Q_{h(m,t)}^{\mu }$ as presented by (5) so that $h(m,t)$ is a function on $m\in (m_0^{\mu },m_{+}^{\mu })$ and $t\in \mathbb{R}\setminus \left\{1\right\}$.

(ii) 

$m_0^{\mu }$ is finite, $h(m,t)=m+(1-t)C_{Q_m^{\mu }}$, and μ is the image by $x\mapsto \sqrt{\sigma }x+m_0^{\mu }$ of the Wigner’s measure (20) with $\sigma >0$ that is sufficiently large.

Proof. 

$\left(i\right)\Rightarrow \left(ii\right)$∀$z\in (\mathbb{B},+\infty )$

$${\mathcal{E}}_{U^{(1,t)}\left(Q_m^{\mu }\right)}\left(z\right)={\mathcal{E}}_{Q_{h(m,t)}^{\mu }}\left(z\right),$$

(22)

where

$${\mathcal{E}}_{\mu }\left(z\right)=z-{\displaystyle \frac{1}{{\mathcal{G}}_{\mu }\left(z\right)}}.$$

(23)

Using (11) together with (22) and knowing (see [13] (Proposition 3.1(iv))) that ${\displaystyle \underset{z⟶+\infty }{lim}{\mathcal{E}}_{\mu }\left(z\right)=m_0^{\mu },}$ we obtain

$$\begin{array}{cc}{\displaystyle h(m,t)=m_0^{Q_{h(m,t)}^{\mu }}}\hfill & {\displaystyle =\underset{z⟶+\infty }{lim}{\mathcal{E}}_{Q_{h(m,t)}^{\mu }}\left(z\right)}\hfill \\ \hspace{1em}\hfill & \hspace{1em}\hspace{1em}\hspace{1em}{\displaystyle =\underset{z⟶+\infty }{lim}{\mathcal{E}}_{U^{(1,t)}\left(Q_m^{\mu }\right)}\left(z\right)=m_0^{U^{(t,1)}\left(Q_m^{\mu }\right)}=m+(1-t)C_{Q_m^{\mu }}.}\hfill \end{array}$$

(24)

From relations (2) and (23), one can see that

$${\mathcal{E}}_{U^{(1,t)}\left(Q_m^{\mu }\right)}\left(z\right)={\mathcal{E}}_{Q_m^{\mu }}\left(z\right)+(1-t)C_{Q_m^{\mu }}.$$

(25)

Combining (22), (24) and (25), we obtain

$${\mathcal{E}}_{Q_{m+(1-t)C_{Q_m^{\mu }}}^{\mu }}\left(z\right)={\mathcal{E}}_{Q_m^{\mu }}\left(z\right)+(1-t)C_{Q_m^{\mu }}.$$

(26)

Lemma 3.3 in [13], say that for $\zeta \in \mathbb{C}\setminus supp\left(\mu \right)$ with $\zeta \ne m+{\mathbb{V}}_{\mu }\left(m\right)/m$, one has

$${\mathcal{E}}_{Q_m^{\mu }}\left(\zeta \right)={\displaystyle \frac{\left(m+\frac{{\mathbb{V}}_{\mu }\left(m\right)}{m}\right){\mathcal{E}}_{\mu }\left(\zeta \right)-m\zeta }{\frac{{\mathbb{V}}_{\mu }\left(m\right)}{m}-(\zeta -{\mathcal{E}}_{\mu }\left(\zeta \right))}}.$$

(27)

Using (27), Equation (26) is

$$\begin{array}{c}{\displaystyle \frac{\left(m+(1-t)C_{Q_m^{\mu }}+\frac{{\mathbb{V}}_{\mu }\left(m+(1-t)C_{Q_m^{\mu }}\right)}{m+(1-t)C_{Q_m^{\mu }}}\right){\mathcal{E}}_{\mu }\left(z\right)-z\left(m+(1-t)C_{Q_m^{\mu }}\right)}{\frac{{\mathbb{V}}_{\mu }\left(m+(1-t)C_{Q_m^{\mu }}\right)}{m+(1-t)C_{Q_m^{\mu }}}-(z-{\mathcal{E}}_{\mu }\left(z\right))}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} ={\displaystyle \frac{\left(m+\frac{{\mathbb{V}}_{\mu }\left(m\right)}{m}\right){\mathcal{E}}_{\mu }\left(z\right)-mz}{\frac{{\mathbb{V}}_{\mu }\left(m\right)}{m}-(z-{\mathcal{E}}_{\mu }\left(z\right))}}+(1-t)C_{Q_m^{\mu }}.\hfill \end{array}$$

(28)

Relation (28) is

$$\begin{array}{c}\left(\left(m+(1-t)C_{Q_m^{\mu }}\right)+{\displaystyle \frac{{\mathbb{V}}_{\mu }\left(m+(1-t)C_{Q_m^{\mu }}\right)}{\left(m+(1-t)C_{Q_m^{\mu }}\right)}}\right){\displaystyle \frac{{\mathbb{V}}_{\mu }\left(m\right)}{m}}{\mathcal{E}}_{\mu }\left(z\right)-z\left(m+(1-t)C_{Q_m^{\mu }}\right){\displaystyle \frac{{\mathbb{V}}_{\mu }\left(m\right)}{m}}\hfill \\ \hspace{1em}\hspace{1em} -\left(\left(m+(1-t)C_{Q_m^{\mu }}\right)+{\displaystyle \frac{{\mathbb{V}}_{\mu }\left(m+(1-t)C_{Q_m^{\mu }}\right)}{\left(m+(1-t)C_{Q_m^{\mu }}\right)}}\right){\mathcal{E}}_{\mu }\left(z\right)(z-{\mathcal{E}}_{\mu }\left(z\right))+(1-t)C_{Q_m^{\mu }}{\mathcal{E}}_{\mu }\left(z\right)(z-{\mathcal{E}}_{\mu }\left(z\right))\\ =\left(m+{\displaystyle \frac{{\mathbb{V}}_{\mu }\left(m\right)}{m}}\right){\displaystyle \frac{{\mathbb{V}}_{\mu }\left(m+(1-t)C_{Q_m^{\mu }}\right)}{\left(m+(1-t)C_{Q_m^{\mu }}\right)}}{\mathcal{E}}_{\mu }\left(z\right)-mz{\displaystyle \frac{{\mathbb{V}}_{\mu }\left(m+(1-t)C_{Q_m^{\mu }}\right)}{\left(m+(1-t)C_{Q_m^{\mu }}\right)}}\\ +(1-t)C_{Q_m^{\mu }}{\displaystyle \frac{{\mathbb{V}}_{\mu }\left(m\right)}{m}}{\displaystyle \frac{{\mathbb{V}}_{\mu }\left(m+(1-t)C_{Q_m^{\mu }}\right)}{\left(m+(1-t)C_{Q_m^{\mu }}\right)}}-(1-t)C_{Q_m^{\mu }}{\displaystyle \frac{{\mathbb{V}}_{\mu }\left(m\right)}{m}}(z-{\mathcal{E}}_{\mu }\left(z\right))\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} -(1-t)C_{Q_m^{\mu }}(z-{\mathcal{E}}_{\mu }\left(z\right)){\displaystyle \frac{{\mathbb{V}}_{\mu }\left(m+(1-t)C_{Q_m^{\mu }}\right)}{\left(m+(1-t)C_{Q_m^{\mu }}\right)}}-\left(m+{\displaystyle \frac{{\mathbb{V}}_{\mu }\left(m\right)}{m}}\right){\mathcal{E}}_{\mu }\left(z\right)(z-{\mathcal{E}}_{\mu }\left(z\right)).\end{array}$$

(29)

From ([12] Section 2), we know that

$$-\infty \le \int x\mu \left(dx\right)=m_0^{\mu }<m_{+}^{\mu }\le supsupp\left(\mu \right)<+\infty .$$

We consider the cases where $-\infty <m_0^{\mu }<+\infty$ and $m_0^{\mu }=-\infty$ separately.

• Assume that $-\infty <m_0^{\mu }<+\infty$. Dividing by z in both side of relation (29) and let z goes to $+\infty$. Recall from ([13], Proposition 3.2) the following

  $${\displaystyle \underset{z⟶+\infty }{lim}\frac{{\mathcal{E}}_{\mu }\left(z\right)}{z}=0\mathrm{and}\underset{z⟶+\infty }{lim}\frac{(z-{\mathcal{E}}_{\mu }\left(z\right)){\mathcal{E}}_{\mu }\left(z\right)}{z}=m_0^{\mu },}$$

  (30)

  and we obtain

  $$\left({\displaystyle \frac{\left(m+(1-t)C_{Q_m^{\mu }}\right)-m_0^{\mu }}{\left(m+(1-t)C_{Q_m^{\mu }}\right)}}\right){\mathbb{V}}_{\mu }\left(m+(1-t)C_{Q_m^{\mu }}\right)={\mathbb{V}}_{\mu }\left(m\right)\left({\displaystyle \frac{m-m_0^{\mu }}{m}}\right).$$

  (31)
  Combining (31) and (4), we obtain

  $${\mathcal{V}}_{\mu }\left(m+(1-t)C_{Q_m^{\mu }}\right)={\mathcal{V}}_{\mu }\left(m\right),\forall m\in (m_0^{\mu },m_{+}^{\mu })\mathrm{and}\forall t\in \mathbb{R}\setminus \left\{1\right\}.$$
  We know that ${\mathcal{V}}_{\mu }(·)\ne 0$ because $\mu$ is non degenerate. Thus, ${\mathcal{V}}_{\mu }(·)=\sigma$ for $\sigma >0$. According to the description of the quadratic VF given in ([11], Theorem 3.2), the measure $\mu$ is the image by $x\mapsto \sqrt{\sigma }x+m_0^{\mu }$ of the Wigner’s measure (20).
• Assume that $m_0^{\mu }=-\infty$. Dividing by $z{\mathcal{E}}_{\mu }\left(z\right)$ on both sides of relation (29), let z go to $+\infty$. Recalling (30), we obtain

  $${\displaystyle \frac{{\mathbb{V}}_{\mu }\left(m+(1-t)C_{Q_m^{\mu }}\right)}{\left(m+(1-t)C_{Q_m^{\mu }}\right)}}={\displaystyle \frac{{\mathbb{V}}_{\mu }\left(m\right)}{m}},\forall m\in (m_0^{\mu },m_{+}^{\mu })\mathrm{and}\forall t\in \mathbb{R}\setminus \left\{1\right\}.$$
  Then, there exists $\gamma \in \mathbb{R}$ such that $\mathbb{V}\left(m\right)=\gamma m$; see ([14], Page 5). That is, $\mathbb{V}\left(m\right)=\kappa m$ cannot be a PVF for ${\mathbb{F}}_{+}\left(\mu \right)$ with $m_0^{\mu }=-\infty$.

We explain the condition made on the parameter $\sigma >0$ so that $h(m,t)=m+(1-t)C_{Q_m^{\mu }}$ is in the domain of means. For this reason, let us observe the first extension (denoted by $\overline{\mathbb{F}}\left(W\right)$ as presented in [15], Section 3.1; see also [14] (Remark 3.2)) of the two-sided CSK family $\mathbb{F}\left(W\right)$ induced by the Wigner’s measure given by (20). We know that $(m_0^{\mu }-\sqrt{\sigma },m_0^{\mu }+\sqrt{\sigma })$ is the two-sided mean domain of $\overline{\mathbb{F}}\left(\mu \right)$, where $\mu =\tau \left(W\right)$ (with $\tau \left(x\right)=\sqrt{\sigma }x+m_0^{\mu }$). The parameter $\sigma$ should be sufficiently large to guarantee that $h(m,t)=m+(1-t)C_{Q_m^{\mu }}$ is in the extended mean domain.

$\left(ii\right)\Rightarrow \left(i\right)$ Supposing that $m_0^{\mu }$ is finite, $h(m,t)=m+(1-t)C_{Q_m^{\mu }}$, and $\mu$ is the image by $x\mapsto \sqrt{\sigma }x+m_0^{\mu }$ of the Wigner’s measure given by (20) for $\sigma >0$. The two-sided mean domain of $\overline{\mathbb{F}}\left(\mu \right)$ is $(m_0^{\mu }-\sqrt{\sigma },m_0^{\mu }+\sqrt{\sigma })$. For $\sigma >0$ that is sufficiently large, one has $m+(1-t)C_{Q_m^{\mu }}\in (m_0^{\mu }-\sqrt{\sigma },m_0^{\mu }+\sqrt{\sigma })$. We have to prove that

$$U^{(1,t)}\left(Q_m^{\mu }\right)=Q_{m+(1-t)C_{Q_m^{\mu }}}^{\mu }.$$

(32)

To achieve (32), the VFs machinery is used. One has

$$m_0^{Q_{m+(1-t)C_{Q_m^{\mu }}}^{\mu }}=m+(1-t)C_{Q_m^{\mu }}=m_0^{U^{(t,1)}\left(Q_m^{\mu }\right)},$$

Then, $\epsilon >0$ exists so that the functions ${\mathcal{V}}_{U^{(1,t)}\left(Q_m^{\mu }\right)}(·)$ and ${\mathcal{V}}_{Q_{m+(1-t)C_{Q_m^{\mu }}}^{\mu }}(·)$ are well defined on $(m+(1-t)C_{Q_m^{\mu }}-\epsilon ,m+(1-t)C_{Q_m^{\mu }}+\epsilon )$. We should demonstrate that

$${\mathcal{V}}_{U^{(1,t)}\left(Q_m^{\mu }\right)}\left(x\right)={\mathcal{V}}_{Q_{m+(1-t)C_{Q_m^{\mu }}}^{\mu }}\left(x\right),\forall x\in (m+(1-t)C_{Q_m^{\mu }}-\epsilon ,m+(1-t)C_{Q_m^{\mu }}+\epsilon )$$

so that (32) follows from (6).

We know from ([14], Equation (3.17)) that ∀$y\ne m$,

$${\mathbb{V}}_{Q_m^{\mu }}\left(y\right)=y\left({\displaystyle \frac{\sigma }{y-m}}+m_0^{\mu }-m\right).$$

(33)

Combining (4) with (33), ∀y in a small neighborhood of m, we obtain

$${\mathcal{V}}_{Q_m^{\mu }}\left(y\right)=\sigma +(m_0^{\mu }-m)(y-m).$$

(34)

Now, ∀$x\in (m+(1-t)C_{Q_m^{\mu }}-\epsilon ,m+(1-t)C_{Q_m^{\mu }}+\epsilon )$, Equations (13) and (34) give

$$\begin{array}{ccc}\hfill {\mathcal{V}}_{U^{(1,t)}\left(Q_m^{\mu }\right)}\left(x\right)& =& {\mathcal{V}}_{Q_m^{\mu }}\left(x+(t-1)C_{Q_m^{\mu }}\right)+(t-1)C_{Q_m^{\mu }}\left(x-m-(1-t)C_{Q_m^{\mu }}\right)\hfill \\ & =& \sigma +\left(m_0^{\mu }-m\left)\right(x-m-(1-t)C_{Q_m^{\mu }}\right)+(t-1)C_{Q_m^{\mu }}\left(x-m-(1-t)C_{Q_m^{\mu }}\right)\hfill \\ & =& \sigma +\left(x-m-(1-t)C_{Q_m^{\mu }}\right)\left(m_0^{\mu }-m-(1-t)C_{Q_m^{\mu }}\right)\hfill \\ & =& {\mathcal{V}}_{Q_{m+(1-t)C_{Q_m^{\mu }}}^{\mu }}\left(x\right).\hfill \end{array}$$

The proof of $\left(i\right)$ is achieved. □

6. Conclusions

In this paper, the notion of the $\mathbb{T}$-transformation is investigated from the viewpoint of CSK families and the corresponding VFs. The formula for VF under the $\mathbb{T}$-transformation is established. This formula is used to give the stability justification of the $\mathcal{FMF}$ under the $(\mathbb{T}=(s,t\left)\right)$-deformation. Furthermore, a new characterization is provided for the semicircle CSK family based on the $(1,t)$-transformation. It is shown that a given CSK family ${\mathbb{F}}_{+}\left(\mu \right)$ is stable with respect to the $U^{(1,t)}$-transformation (i.e., $U^{(1,t)}\left({\mathbb{F}}_{+}\left(\mu \right)\right)={\mathbb{F}}_{+}\left(\mu \right)$) if and only if $\mu$ is of the Wigner’s semicircle-type measure. These works deepen our understanding of the $(s,t)$-transformation in a noncommutative probability setting.