V_a-Deformed Free Convolution

Abstract

In this article, we study $V_a$-transformation of a measure and of a convolution (denoted by $\boxed{a}$) defined for $a\in \mathbb{R}$. We provide significant insights into the stability of the free Meixner family of probability measures under $V_a$-transformation. We show that the $V_a$-transformation of measures (of convolutions) of any member of the free Meixner family remains in the free Meixner family. We also present some properties of the Marchenko–Pastur law in connection with $\boxed{a}$-convolution. In addition, some new limit theorems are proved for the $\boxed{a}$-convolution incorporating both free and Boolean additive convolutions. Furthermore, some properties related to $V_a$-deformed free cumulants are presented.

1. Introduction

A key idea in probability theory and statistics is the transformation of probability measures and convolutions, which investigates how different mathematical operations can change or control probability measures. Understanding the relationships between several random variables and how they can be converted into new random variables requires knowledge of this subject. A strong toolkit for examining and working with random variables is provided by the transformation of probability measures. Data scientists and statisticians can more effectively describe complicated events, evaluate data, and make defensible conclusions based on probabilistic reasoning by knowing how to convert one measure into another. This subject bridges the gap between theoretical probability and real-world applications in a number of fields, such as the social sciences, engineering, and finance. This topic has been explored and expanded in many ways in a number of articles, including [1,2,3,4,5,6,7,8]. In [9], a novel transformation of measures is presented in the framework of free probability. For the Cauchy–Stieltjes transformation of a probability measure $\rho$ with finite second moment, deformation is taken into consideration by supplying a certain (fixed) proportion of the variance of $\rho$ to its reciprocal. The derived function is the reciprocal of the Cauchy–Stieltjes transform of a certain probability measure denoted as $V_a\left(\rho \right)$ and called $V_a$-transformation of measure $\rho$, as elucidated in [9]. For $a\in \mathbb{R}$, the following expression is relevant:

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

(1)

where ${\mathbf{r}}_2\left(\rho \right)$ is the variance (second free cumulant) of $\rho$ and

$${\mathbf{G}}_{\rho }\left(w\right)=\int {\displaystyle \frac{1}{w-\zeta }}\rho \left(d\zeta \right),w\in \mathbb{C}\setminus supp\left(\rho \right)$$

(2)

is the Cauchy–Stieltjes transform of $\rho$.

Based on the $V_a$-transformation of measures, a novel type of convolution, referred to as the $V_a$-transformation of free convolution and indicated by $\begin{array}{|c|}\hline a\\ \hline\end{array}$-convolution, is defined in [9] as follows: let $\rho$ and $\varrho$ be probability measures possessing finite second moments. Their $V_a$-transformed free convolution is defined as follows:

$$\rho \begin{array}{|c|}\hline a\\ \hline\end{array}\varrho =V_{-a}(V_a\left(\rho \right)⊞V_a\left(\varrho \right))$$

(3)

where ⊞ is the free additive convolution.

The central limit theorem with respect to $\begin{array}{|c|}\hline a\\ \hline\end{array}$-convolution is established in [9], with the limiting measure being the standard Wigner distribution. The Poisson-type limit theorem related to $\begin{array}{|c|}\hline a\\ \hline\end{array}$-convolution is also established, with the resulting limiting measure referred to as the $V_a$-free Poisson distribution. In [10], the concept of $\begin{array}{|c|}\hline a\\ \hline\end{array}$-convolution is examined through the Cauchy–Stieltjes Kernel families and their associated variance functions. A formula is presented for the variance function when considering a power of $\begin{array}{|c|}\hline a\\ \hline\end{array}$-convolution of the generating measure. Additionally, an approximation is provided for elements within the $V_a$-free Poisson Cauchy–Stieltjes Kernel family.

However, in [9], the concept of conditionally free convolution is investigated, which is a specific type of convolution operation in free probability theory. The connections between free convolution (introduced by Dan Voiculescu) and conditionally free convolution is explored, focusing on the algebraic and probabilistic structures arising from these convolutions. It was shown how associative convolutions can be derived from conditionally free convolutions. A significant contribution to the theory of free probability is given by introducing and studying associative convolutions arising from conditionally free convolution. These results deepen the understanding of convolutions in non-commutative probability and open up new possibilities for analyzing complex probabilistic structures involving random variables with conditional dependencies. Furthers works related to conditionally free convolution can be found in [11,12,13,14].

This article is a continuation of the study of $\begin{array}{|c|}\hline a\\ \hline\end{array}$-convolution within the context of Cauchy–Stieltjes Kernel families. For a better presentation of the article’s main results, we need first to introduce some facts about Cauchy–Stieltjes Kernel families, which concerns families of probabilities defined in a manner analogous to that of natural exponential families by incorporating the Cauchy–Stieltjes Kernel ${\displaystyle \frac{1}{1-\vartheta \zeta }}$ in the place of the exponential kernel $exp\left(\vartheta \zeta \right)$. The Cauchy–Stieltjes Kernel families were studied in [15] for measures having compact support. Further findings are proved in [16] by involving measures having one-sided support boundary from above. ${\mathcal{P}}_{ba}$ denotes the set of (non-degenerate) probability measures with one-sided support boundary from above. Let $\rho \in {\mathcal{P}}_{ba}$, then with ${\displaystyle \frac{1}{{\vartheta }_{+}^{\rho }}}=max\{0,sup\mathrm{supp}\left(\rho \right)\}$, the transform

$${\mathbf{M}}_{\rho }\left(\vartheta \right)=\int {\displaystyle \frac{1}{1-\vartheta \zeta }}\rho \left(d\zeta \right)$$

is finite ∀ $\vartheta \in [0,{\vartheta }_{+}^{\rho })$. The Cauchy–Stieltjes Kernel family induced by $\rho$ is the set of probabilities as follows:

$${\mathbf{F}}_{+}\left(\rho \right)=\{{\mathbf{P}}_{\vartheta }^{\rho }\left(d\zeta \right)={\displaystyle \frac{1}{{\mathbf{M}}_{\rho }\left(\vartheta \right)(1-\vartheta \zeta )}}\rho \left(d\zeta \right):\vartheta \in (0,{\vartheta }_{+}^{\rho })\}.$$

Following [16], the map $\vartheta \mapsto {\mathbf{K}}_{\rho }\left(\vartheta \right)=\int \zeta {\mathbf{P}}_{\vartheta }^{\rho }\left(d\zeta \right)$ is increasing strictly on $(0,{\vartheta }_{+}^{\rho })$. The mean domain of ${\mathbf{F}}_{+}\left(\rho \right)$ is the interval $(m_1^{\rho },m_{+}^{\rho })={\mathbf{K}}_{\rho }\left((0,{\vartheta }_{+}^{\rho })\right)$. Mean parametrization is then provided for ${\mathbf{F}}_{+}\left(\rho \right)$: the inverse of ${\mathbf{K}}_{\rho }(·)$ is denoted by ${\Phi }_{\rho }(·)$. For $m\in (m_1^{\rho },m_{+}^{\rho })$, consider ${\mathbf{Q}}_m^{\rho }\left(d\zeta \right)={\mathbf{P}}_{{\Phi }_{\rho }\left(m\right)}^{\rho }\left(d\zeta \right)$. Then,

$${\mathbf{F}}_{+}\left(\rho \right)=\{{\mathbf{Q}}_m^{\rho }\left(d\zeta \right):m\in (m_1^{\rho },m_{+}^{\rho })\}.$$

From [16], we have

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

where

$$\mathbb{B}=\mathbb{B}\left(\rho \right)=max\{0,sup\mathrm{supp}\left(\rho \right)\}={\displaystyle \frac{1}{{\vartheta }_{+}^{\rho }}}\in [0,\infty ).$$

If $\rho$ has one-sided support boundary from below, the corresponding Cauchy–Stieltjes Kernel family is represented by the notation ${\mathbf{F}}_{-}\left(\rho \right)$ and $\vartheta \in ({\vartheta }_{-}^{\rho },0)$, where ${\vartheta }_{-}^{\rho }$ is either $1/\mathbb{A}\left(\rho \right)$ or $-\infty$ with $\mathbb{A}=\mathbb{A}\left(\rho \right)=min\{infsupp(\rho ),0\}$. The interval $(m_{-}^{\rho },m_1^{\rho })$ is the mean domain for ${\mathbf{F}}_{-}\left(\rho \right)$ with $m_{-}^{\rho }=\mathbb{A}-1/{\mathbf{G}}_{\rho }\left(\mathbb{A}\right)$. If the support of $\rho$ is compact, then the two-sided Cauchy–Stieltjes Kernel (CSK) family is $\mathbf{F}\left(\rho \right)={\mathbf{F}}_{+}\left(\rho \right)\cup {\mathbf{F}}_{-}\left(\rho \right)\cup \left\{\rho \right\}$.

The variance function (VF) is defined by (see [15])

$$m\mapsto {\mathfrak{V}}_{\rho }\left(m\right)=\int {(\zeta -m)}^2{\mathbf{Q}}_m^{\rho }\left(d\zeta \right).$$

If $\rho \in {\mathcal{P}}_{ba}$ does not have the first moment, then every element of ${\mathbf{F}}_{+}\left(\rho \right)$ has infinite variance. In [16], a new concept, called the pseudo-variance function, denoted as ${\mathbb{V}}_{\rho }(·)$, is introduced as follows:

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

If $m_1^{\rho }=\int \zeta \rho \left(d\zeta \right)$ is finite, then the VF exists, as shown in [16], and

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

(4)

Now, we recall the definition of $V_a$-deformed free cumulant transform, and we present the goal of this paper in more detail. ${\mathcal{P}}_c$ denotes the set of (non-degenerate) compactly supported probabilities on $\mathbb{R}$. For $\varrho \in {\mathcal{P}}_c$, the ${\mathfrak{R}}^a$-transform is considered as follows:

$${\mathfrak{R}}_{\varrho }^a\left(z\right):={\mathfrak{R}}_{V_a\left(\varrho \right)}\left(z\right),$$

(5)

where the free cumulant transformation, ${\mathfrak{R}}_{\lambda }$, of $\lambda \in {\mathcal{P}}_c$ is defined as follows:

$${\mathfrak{R}}_{\lambda }\left({\mathbf{G}}_{\lambda }\left(\xi \right)\right)=\xi -{\displaystyle \frac{1}{{\mathbf{G}}_{\lambda }\left(\xi \right)}},\forall \xi \mathrm{close}\hspace{0.17em}\mathrm{to}\infty .$$

For $\rho$ and $\varrho \in {\mathcal{P}}_c$, we have

$${\mathfrak{R}}_{\rho \begin{array}{|c|}\hline a\\ \hline\end{array}\varrho }^a\left(z\right)={\mathfrak{R}}_{\rho }^a\left(z\right)+{\mathfrak{R}}_{\varrho }^a\left(z\right).$$

(6)

A measure $\varrho \in {\mathcal{P}}_c$ is $\begin{array}{|c|}\hline a\\ \hline\end{array}$-infinitely divisible if, for every $p\in \mathbb{N}$, there exists ${\varrho }_p\in {\mathcal{P}}_c$ so that

$$\varrho =\underset{p\mathrm{times}}{\underbrace{{\varrho }_p\begin{array}{|c|}\hline a\\ \hline\end{array}\cdots \begin{array}{|c|}\hline a\\ \hline\end{array}{\varrho }_p}}.$$

The $\gamma$-fold $\begin{array}{|c|}\hline a\\ \hline\end{array}$-convolution of $\varrho \in {\mathcal{P}}_c$ with itself is represented as ${\varrho }^{\begin{array}{|c|}\hline a\\ \hline\end{array}\gamma }$. For every real $\gamma \ge 1$, this operation is well-defined (see [17]), and we have

$${{\mathfrak{R}}^a}_{{\varrho }^{\begin{array}{|c|}\hline a\\ \hline\end{array}\gamma }}\left(z\right)=\gamma {\mathfrak{R}}_{\varrho }^a\left(z\right).$$

(7)

The objectives in this paper related to the $\begin{array}{|c|}\hline a\\ \hline\end{array}$-convolution are as follows: in Section 2, we provide significant insights into the stability of the free Meixner family of probability measures under $V_a$-transformation. We show that the $V_a$-transformation of measures (of convolutions) of any member of the free Meixner family remains in the free Meixner family. In addition, some properties related to the Marchenko–Pastur law are presented in Section 3: Let $\mathbf{F}\left(\rho \right)=\{{\mathbf{Q}}_m^{\rho }\left(d\zeta \right);m\in (m_{-}^{\rho },m_{+}^{\rho })\}$ be the Cauchy–Stieltjes Kernel (CSK) family induced by $\rho \in {\mathcal{P}}_c$. For $\gamma >1$, the set of measures is introduced as follows:

$${\left(\mathbf{F}\left(\rho \right)\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\gamma }=\left\{{\left({\mathbf{Q}}_m^{\rho }\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\gamma }\left(d\zeta \right):m\in (m_{-}^{\rho },m_{+}^{\rho })\right\}.$$

(8)

The set of measures ${\left(\mathbf{F}\left(\rho \right)\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\gamma }$ is the family containing powers of $\begin{array}{|c|}\hline a\\ \hline\end{array}$-convolution of measure ${\mathbf{Q}}_m^{\rho }$. The question is whether the family ${\left(\mathbf{F}\left(\rho \right)\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\gamma }$ is a re-parametrization of the CSK family $\mathbf{F}\left(\rho \right)$. In other words, measure $\rho$ must be determined so that

$${\left(\mathbf{F}\left(\rho \right)\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\gamma }=\mathbf{F}\left(\rho \right)?$$

(9)

If relation (9) holds, then $\rho$ is of the Marchenko–Pastur type measure.

Furthermore, $\varrho \in {\mathcal{P}}_c$ is considered, and the set of measures is introduced as follows:

$$\Omega =\mathbf{F}\left(\rho \right)\begin{array}{|c|}\hline a\\ \hline\end{array}\mathbf{F}\left(\varrho \right)=\{{\mathbf{Q}}_{m_1}^{\rho }\begin{array}{|c|}\hline a\\ \hline\end{array}{\mathbf{Q}}_{m_2}^{\varrho }:m_1\in (m_{-}^{\rho },m_{+}^{\rho })\mathrm{and}{m}_2\in (m_{-}^{\varrho },m_{+}^{\varrho })\}.$$

(10)

We prove that if $\Omega$ remains a CSK family (i.e., $\Omega =\mathbf{F}\left(\lambda \right)$ for some $\lambda \in {\mathcal{P}}_c$), then $\lambda$, $\rho$, and $\varrho$ are of the Marchenko–Pastur type law up to affinity.

In Section 4, some new limit theorems are proved for the $\begin{array}{|c|}\hline a\\ \hline\end{array}$-convolution based on the VF and incorporating both free and Boolean additive convolutions. Finally, in Section 5, some properties related to $V_a$-deformed free cumulants are presented in connection with the concept of VF.

To close this section, some useful facts are presented in the following remark to help in the proof of this paper’s results.

Remark 1.

(i) Let $\rho \in {\mathcal{P}}_{ba}$. We know that ${\mathbb{V}}_{\rho }(·)$ determines ρ, as shown in Equation (3.9) in [16]. $\Delta =\Delta \left(m\right)=m+{\displaystyle \frac{{\mathbb{V}}_{\rho }\left(m\right)}{m}}$ is considered, then

$${\mathbb{G}}_{\rho }(\Delta )={\displaystyle \frac{m}{{\mathbb{V}}_{\rho }\left(m\right)}}.$$

If $m_1^{\rho }$ is finite, then

$${\mathbb{G}}_{\rho }(\Delta )={\displaystyle \frac{m-m_1^{\rho }}{{\mathfrak{V}}_{\rho }\left(m\right)}}.$$

(11)

Thus, ${\mathfrak{V}}_{\rho }(·)$ and $m_1^{\rho }$ characterizes ρ.

(ii) Let ${\chi }_{(\kappa ,\beta )}\left(\rho \right)$ denote the image of ρ by ${\chi }_{(\kappa ,\beta )}:\zeta \mapsto \kappa \zeta +\beta$, where $\kappa \ne 0$ and $\beta \in \mathbb{R}$. Then, ∀ m close sufficiently to $m_1^{{\chi }_{(\kappa ,\beta )}\left(\rho \right)}={\chi }_{(\kappa ,\beta )}\left(m_1^{\rho }\right)=\kappa m_1^{\rho }+\beta$,

$${\mathbb{V}}_{{\chi }_{(\kappa ,\beta )}\left(\rho \right)}\left(m\right)={\displaystyle \frac{{\kappa }^{2}m}{m-\beta }}{\mathbb{V}}_{\rho }\left({\displaystyle \frac{m-\beta }{\kappa }}\right).$$

If the VF exists,

$${\mathfrak{V}}_{{\chi }_{(\kappa ,\beta )}\left(\rho \right)}\left(m\right)={\kappa }^2{\mathfrak{V}}_{\rho }\left({\displaystyle \frac{m-\beta }{\kappa }}\right).$$

(12)

Furthermore, ∀ ξ is sufficiently close to 0 as follows:

$${\mathfrak{R}}_{{\chi }_{(\kappa ,\beta )}\left(\rho \right)}\left(\xi \right)=\kappa {\mathfrak{R}}_{\rho }\left(\kappa \xi \right)+\beta .$$

(13)

(iii) Let $\rho \in {\mathcal{P}}_{ba}$ with the finite second moment. We know from [10] that ∀ $a\in \mathbb{R}$ and ∀ m are sufficiently close to $m_1^{V_a\left(\nu \right)}=m_1^{\nu }-a{\mathit{r}}_2\left(\nu \right)$ as follows:

$${\mathfrak{V}}_{V_a\left(\nu \right)}\left(m\right)={\mathfrak{V}}_{\nu }(m+a{\mathit{r}}_2\left(\nu \right))+a{\mathit{r}}_2\left(\nu \right)(m+a{\mathit{r}}_2\left(\nu \right)-m_1^{\nu }).$$

(14)

In addition, ∀ m close sufficiently to $m_1^{{\nu }^{\begin{array}{|c|}\hline a\\ \hline\end{array}\alpha }}=\alpha m_1^{\nu }$,

$${\mathfrak{V}}_{{\nu }^{\begin{array}{|c|}\hline a\\ \hline\end{array}\alpha }}\left(m\right)=\alpha {\mathfrak{V}}_{\nu }(m/\alpha )+a{\mathit{r}}_2\left(\nu \right)(1-\alpha )(m-\alpha m_1^{\nu }).$$

(15)

2. Remarks on the Free Meixner Family of Measures

Bryc [15] described the class of quadratic CSK families having VF of the following form:

$${\mathcal{V}}_{\mu }\left(m\right)=1+a_{1}m+a_2{m}^2,a_1\in \mathbb{R},a_2\ge -1$$

(16)

with $m_1^{\mu }=0$. The corresponding probability measures are the so-called free Meixner family of measures (which we denote by $\mathbf{FMF}$):

$$\mu \left(d\zeta \right)={\displaystyle \frac{\sqrt{4(1+a_2)-{(\zeta -a_1)}^2}}{2\pi (a_2{\zeta }^2+a_1\zeta +1)}}{1}_{(a_1-2\sqrt{1+a_2},a_1+2\sqrt{1+a_2})}\left(\zeta \right)d\zeta +p_1{\delta }_{{\zeta }_1}+p_2{\delta }_{{\zeta }_2}.$$

(17)

The discrete part of $\mu$ is absent, except for the following cases:

(i)

If $a_2=0,a_1^2>1$, then $p_1=1-1/a_1^2,x_1=-1/a_1,p_2=0$.

(ii)

If $a_2>0$ and $a_1^2>4a_2$, then $p_1=max\{0,1-{\displaystyle \frac{|a_1|-\sqrt{a_1^2-4a_2}}{2a_2\sqrt{a_1^2-4a_2}}}\},p_2=0$, and ${\zeta }_1=\pm {\displaystyle \frac{|a_1|-\sqrt{a_1^2-4a_2}}{2a_2}}$ with the sign opposite to the sign of a.

(iii)

If $-1\le a_2<0$, then there are two atoms at

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

These results covers a number of important measures that appeared in the literature. Up to dilation and convolution with a degenerate law ${\delta }_{a_1}$, measure $\mu$ is calculated as follows:

(i)

Wigner’s semicircle (free Gaussian) law if $a_1=a_2=0$.

(ii)

The Marchenko–Pastur (free Poisson) type law if $a_2=0$ and $a_1\ne 0$.

(iii)

The free Pascal (free negative binomial) type law if $a_2>0$ and $a_1^2>4a_2$.

(iv)

The free Gamma type law if $a_2>0$ and $a_1^2=4a_2$.

(v)

The free analog of hyperbolic type law if $a_2>0$ and $a_1^2<4a_2$.

(vi)

The free binomial type law if $-1\le a_2<0$.

Now, we show that the free Meixner family remains invariant under the $V_a$-transformation of measures.

Theorem 1.

If $\nu \in \mathbf{FMF}$, then for every $a\in \mathbb{R}$, ${\mathbf{T}}_{a{\mathit{r}}_2\left(\nu \right)}\left(V_a\left(\nu \right)\right)\in \mathbf{FMF}$, with ${\mathbf{T}}_{\beta }\left(x\right)={\chi }_{(1,\beta )}\left(x\right)=x+\beta$, for $\beta \in \mathbb{R}$.

Proof. 

Suppose that $\nu \in \mathbf{FMF}$. Then, the VF of $\mathcal{K}\left(\nu \right)$ may be written as

$${\mathcal{V}}_{\nu }\left(m\right)=1+a_1^{\prime }m+a_2^{\prime }{m}^2,a_1^{\prime }\in \mathbb{R},a_2^{\prime }\ge -1,$$

(18)

with $m_1^{\nu }=0$ and ${\mathbf{r}}_2\left(\nu \right)=1$. Combining (18), (14), and (12), ∀ m close to $m_1^{{\mathbf{T}}_a\left(V_a\left(\nu \right)\right)}=0$, we obtain

$${\mathcal{V}}_{{\mathbf{T}}_a\left(V_a\left(\nu \right)\right)}\left(m\right)=1+(a_1^{\prime }+a)m+a_2^{\prime }{m}^2,$$

(19)

which is a quadratic VF in the mean m. Then, ${\mathbf{T}}_a\left(V_a\left(\nu \right)\right)$ belongs to $\mathbf{FMF}$. □

In the following, using some particular examples of measures, we illustrate the importance of Theorem 1.

Corollary 1.

Consider the symmetric Bernoulli law $\nu \left(d\zeta \right)={\displaystyle \frac{1}{2}}{\delta }_{-1}+{\displaystyle \frac{1}{2}}{\delta }_1$. Then, ${\mathbf{T}}_a\left(V_a\left(\nu \right)\right)$ is the free binomial measure with $a_1=a$ and $a_2=-1$.

Proof. 

We know that the VF of the CSK family induced by the symmetric Bernoulli law is provided by (18) for $a_1^{\prime }=0$ and $a_2^{\prime }=-1$. From (19), we have that ${\mathcal{V}}_{{\mathbf{T}}_a\left(V_a\left(\nu \right)\right)}\left(m\right)=1+am-m^2$. This end the proof by the use of (11). □

Corollary 2.

For $a_1^{\prime }\ne 0$ and $a_2^{\prime }=0$, consider the Marchenko–Pastur measure

$$\nu \left(d\zeta \right)={\displaystyle \frac{\sqrt{4-{(\zeta -a_1^{\prime })}^2}}{2\pi (a_1^{\prime }\zeta +1)}}{1}_{(a_1^{\prime }-2,a_1^{\prime }+2)}\left(\zeta \right)d\zeta +{(1-1/a_1^{\prime 2})}^{+}{\delta }_{-1/a_1^{\prime }}$$

(20)

with mean $m_1^{\nu }=0$ and variance 1. Then, ${\mathbf{T}}_a\left(V_a\left(\nu \right)\right)$ is the Marchenko–Pastur measure with $a_1=a_1^{\prime }+a$ and $a_2=a_2^{\prime }=0$.

Proof. 

The proof follows by comparing $a_1^{\prime }\ne 0$ and $a_2^{\prime }=0$, Relation (19) with Relation (16). The conclusion is obtained from the fact that the VF together with the first moment determine the generating measure as presented in (11). □

Now, we show that the free Meixner family remains invariant under the power of $\begin{array}{|c|}\hline a\\ \hline\end{array}$-convolution.

Theorem 2.

If $\nu \in \mathbf{FMF}$, then for every $\alpha >0$ so that ${\nu }^{\begin{array}{|c|}\hline a\\ \hline\end{array}\alpha }$ is well defined, we have $D_{1/\sqrt{\alpha }}\left({\nu }^{\begin{array}{|c|}\hline a\\ \hline\end{array}\alpha }\right)\in \mathbf{FMF}$.

Proof. 

Suppose that $\nu \in \mathbf{FMF}$. Combining (18), (15), and (12), ∀ m are close to $m_1^{D_{1/\sqrt{\alpha }}\left({\nu }^{\begin{array}{|c|}\hline a\\ \hline\end{array}\alpha }\right)}=0$, we obtain

$${\mathcal{V}}_{D_{1/\sqrt{\alpha }}\left({\nu }^{\begin{array}{|c|}\hline a\\ \hline\end{array}\alpha }\right)}\left(m\right)=1+{\displaystyle \frac{a_1^{\prime }+a(1-\alpha )}{\sqrt{\alpha }}}m+{\displaystyle \frac{a_2^{\prime }}{\alpha }}{m}^2,$$

(21)

which is a quadratic VF in the mean m. Then, $D_{1/\sqrt{\alpha }}\left({\nu }^{\begin{array}{|c|}\hline a\\ \hline\end{array}\alpha }\right)$ belongs to $\mathbf{FMF}$. □

Corollary 3.

Let ν be the Marchenko–Pastur measure (20). Then, $D_{1/\sqrt{\alpha }}\left({\nu }^{\begin{array}{|c|}\hline a\\ \hline\end{array}\alpha }\right)$ is the Marchenko–Pastur measure with $a_1={\displaystyle \frac{a_1^{\prime }+a(1-\alpha )}{\sqrt{\alpha }}}$ and $a_2=a_2^{\prime }=0$.

3. Notes on the Marchenko–Pastur Law Based on $\begin{array}{|c|}\hline \mathit{a}\\ \hline\end{array}$-Convolution

This section is devoted to state and demonstrate some properties related to the Marchenko–Pastur law in connection with the $\begin{array}{|c|}\hline a\\ \hline\end{array}$-convolution. The Marchenko–Pastur measure considered in this section is given, for $a_1\ne 0$, by

$${\mathbf{MP}}_{a_1}\left(d\zeta \right)={\displaystyle \frac{\sqrt{\left({(a_1+1)}^2-\zeta \right)\left(\zeta -{(a_1-1)}^2\right)}}{2\pi a_1^2\zeta }}{\mathbf{1}}_{\left({(a_1-1)}^2,{(a_1+1)}^2\right)}\left(\zeta \right)d\zeta +{(1-1/a_1^2)}^{+}{\delta }_0$$

(22)

with $m_1^{{\mathbf{MP}}_{a_1}}=1$. It represents the image of (17) (for $a_1\ne 0$ and $a_2=0$) by the map ${\chi }_{(1,a_1)}\left(x\right)=1+a_{1}x$. We have

$${\mathbb{V}}_{{\mathbf{MP}}_{a_1}}\left(m\right)={\displaystyle \frac{a_1^2{m}^2}{m-1}},$$

(23)

and

$$(m_{-}^{{\mathbf{MP}}_{a_1}},m_{+}^{{\mathbf{MP}}_{a_1}})=\left\{\begin{array}{cc}(1-|a_1|,1+|a_1\left|\right),\hfill & \mathrm{if}{a}_1^2\le 1;\hfill \\ (0,1+|a_1\left|\right),\hfill & \mathrm{if}{a}_1^2>1.\hfill \end{array}\right.$$

(24)

See Section 3 in [18].

Theorem 3.

Let $\rho \in {\mathcal{P}}_c$. For $\gamma >1$, consider the set of probabilities

$${\left(\mathbf{F}\left(\rho \right)\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\gamma }=\left\{{\left({\mathbf{Q}}_m^{\rho }\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\gamma }\left(d\zeta \right):m\in (m_{-}^{\rho },m_{+}^{\rho })\right\}.$$

(25)

If ${\left(\mathbf{F}\left(\rho \right)\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\gamma }$ is a re-parametrization of $\mathbf{F}\left(\rho \right)$, then ρ is the Marchenko–Pastur measure (22), up to a scale transformation, with $a_1^2>1$ such that $|a_1|$ is sufficiently large.

Proof. 

Suppose that ${\left(\mathbf{F}\left(\rho \right)\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\gamma }=\mathbf{F}\left(\rho \right)$. Then, ∀ $t\in (m_{-}^{\rho },m_{+}^{\rho })$, there exists $s\in (m_{-}^{\rho },m_{+}^{\rho })$ so that ${\left({\mathbf{Q}}_t^{\rho }\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\gamma }={\mathbf{Q}}_s^{\rho }$. That is, ∀ z close to 0

$$\gamma {{\mathfrak{R}}^a}_{{\mathbf{Q}}_t^{\rho }}\left(z\right)={{\mathfrak{R}}^a}_{{\mathbf{Q}}_s^{\rho }}\left(z\right).$$

(26)

Let $z\to 0$ in both side of (26). Based on Proposition 3.4 (iii) in [10], we obtain $\gamma m_1^{V_a\left({\mathbf{Q}}_t^{\rho }\right)}=m_1^{V_a\left({\mathbf{Q}}_s^{\rho }\right)}$.

$$\gamma [t-a{\mathfrak{V}}_{\rho }\left(t\right)]=s-a{\mathfrak{V}}_{\rho }\left(s\right).$$

(27)

See Theorem 3.2 in [10]. Relation (27) states that $s=\gamma t$ and then ${\mathfrak{V}}_{\rho }\left(\gamma t\right)=\gamma {\mathfrak{V}}_{\rho }\left(t\right)$, ∀ $t\in (m_{-}^{\rho },m_{+}^{\rho })$ and ∀ $\gamma >1$. We know that ${\mathfrak{V}}_{\rho }(·)\ne 0$ as $\rho$ is (by assumption) non-degenerate. Then, ${\mathfrak{V}}_{\rho }\left(t\right)=\eta t$, for $\eta >0$.

• If $m_1^{\rho }=0$, then there is no VF of the form $V\left(t\right)=\eta t$ with $\eta >0$, as shown in page 6 of [18].
• If $m_1^{\rho }\ne 0$, then $\rho$ is the image by $\zeta \mapsto m_1^{\rho }\zeta$ of the Marchenko–Pastur measure as provided by (22). In this case, $\eta =a_1^2{m}_1^{\rho }$.

Remark 2.

The choice of the parameter $a_1\ne 0$ of ${\mathit{MP}}_{a_1}$ is justified in this remark. For $t\in (m_{-}^{\rho },m_{+}^{\rho })$, we must have $\gamma t\in (m_{-}^{\rho },m_{+}^{\rho })$. The law ρ is the image of ${\mathit{MP}}_{a_1}$ by $:x\mapsto m_1^{\rho }x$. If $m_1^{\rho }>0$, Combining Theorem 3.4 in [19] with (24), we obtain $(m_{-}^{\rho },m_{+}^{\rho })=(m_1^{\rho }(1-|a_1\left|\right),m_1^{\rho }(1+|a_1\left|\right)).$ If $m_1^{\rho }<0$, $(m_{-}^{\rho },m_{+}^{\rho })=(m_1^{\rho }(1+|a_1\left|\right),m_1^{\rho }(1-|a_1\left|\right))$ is the domain of means. In all cases, we should have $a_1^2>1$ such that $|a_1|$ is large enough to ensure that $\gamma t$ is in the mean domain.

We now demonstrate that, in Theorem 3, the inverse implication is invalid.

$${\left({\mathbf{Q}}_t^{\rho }\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\gamma }\ne {\mathbf{Q}}_{\gamma t}^{\rho }.$$

(28)

We have $m_1^{{\left({\mathbf{Q}}_t^{\rho }\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\gamma }}=\gamma t=m_1^{{\mathbf{Q}}_{\gamma t}^{\rho }}$. Then, $\varsigma >0$ exists so that ${\mathfrak{V}}_{{\left({\mathbf{Q}}_t^{\rho }\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\gamma }}(·)$ and ${\mathfrak{V}}_{{\mathbf{Q}}_{\gamma t}^{\rho }}(·)$ are well defined on $(\gamma t,\gamma t+\varsigma )$. From [18], we know that

$${\mathbb{V}}_{Q_m^{\rho }}\left(y\right)=y^2\left({\displaystyle \frac{a^2{m}_1^{\rho }}{y-m}}+{\displaystyle \frac{m_1^{\rho }}{m}}-1\right),y\ne m.$$

This, together with (4), implies that

$${\mathfrak{V}}_{{\mathbf{Q}}_m^{\rho }}\left(y\right)=a^{2}y{m}_1^{\rho }+y(y-m)\left[{\displaystyle \frac{m_1^{\rho }}{m}}-1\right],y\ne m.$$

(29)

Using (15) and (29), ∀ $x\in (\gamma t,\gamma t+\varsigma )$, we obtain

$$\begin{array}{ccc}\hfill {\mathfrak{V}}_{{\left({\mathbf{Q}}_t^{\rho }\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\gamma }}\left(x\right)& =& \gamma {\mathfrak{V}}_{{\mathbf{Q}}_t^{\rho }}(x/\gamma )+a{r}_2\left({\mathbf{Q}}_t^{\rho }\right)(1-\gamma )(x-\gamma m_1^{{\mathbf{Q}}_t^{\rho }})\hfill \\ & =& a^{2}x{m}_1^{\rho }+x(x/\gamma -t)(m_1^{\rho }/t-1)+a{\mathfrak{V}}_{\rho }\left(t\right)(1-\gamma )(x-\gamma t)\hfill \\ & =& a^{2}x{m}_1^{\rho }+x(x/\gamma -t)(m_1^{\rho }/t-1)+a^3{m}_1^{\rho }{t}^2(1-\gamma )(x-\gamma t)\hfill \\ & \ne & {\mathfrak{V}}_{{\mathbf{Q}}_{\gamma t}^{\rho }}\left(x\right)=a^{2}x{m}_1^{\rho }+x(x-\gamma t)(m_1^{\rho }/\left(\gamma t\right)-1).\hfill \end{array}$$

This ends the proof of (28), by using (11). □

The following result is also related to the Marchenko–Pastur law.

Theorem 4.

Let ρ and $\varrho \in {\mathcal{P}}_c$. Introduce the set of measures

$$\Omega =\mathbf{F}\left(\rho \right)\begin{array}{|c|}\hline a\\ \hline\end{array}\mathbf{F}\left(\varrho \right)=\{{\mathbf{Q}}_{m_1}^{\rho }\begin{array}{|c|}\hline a\\ \hline\end{array}{\mathbf{Q}}_{m_2}^{\varrho }:m_1\in (m_{-}^{\rho },m_{+}^{\rho })and{m}_2\in (m_{-}^{\varrho },m_{+}^{\varrho })\}.$$

(30)

If Ω remains a CSK family (i.e., $\Omega =\mathbf{F}\left(\tau \right)$ for some $\tau \in {\mathcal{P}}_c$), then τ, ρ, and ϱ are of the Marchenko–Pastur measure up to affinity.

Proof. 

Suppose that $\Omega =\mathbf{F}\left(\tau \right)$ for some $\tau \in {\mathcal{P}}_c$ (we may suppose, without loss of generality, that $0\in (m_{-}^{\tau },m_{+}^{\tau })$). Then, ∀ $m_1\in (m_{-}^{\rho },m_{+}^{\rho })$ and ∀ $m_2\in (m_{-}^{\varrho },m_{+}^{\varrho })$, and there is $r\in (m_{-}^{\tau },m_{+}^{\tau })$ so that ${\mathbf{Q}}_r^{\tau }={\mathbf{Q}}_{m_1}^{\rho }\begin{array}{|c|}\hline a\\ \hline\end{array}{\mathbf{Q}}_{m_2}^{\varrho }$. ∀ z close to 0 as follows:

$${{\mathfrak{R}}^a}_{{\mathbf{Q}}_r^{\tau }}\left(z\right)={{\mathfrak{R}}^a}_{{\mathbf{Q}}_{m_1}^{\rho }}\left(z\right)+{{\mathfrak{R}}^a}_{{\mathbf{Q}}_{m_2}^{\varrho }}\left(z\right).$$

(31)

Let $z\to 0$ on both sides of (31). Based on Proposition 3.4 (iii) in [10], we obtain $m_1^{V_a\left({\mathbf{Q}}_r^{\tau }\right)}=m_1^{V_a\left({\mathbf{Q}}_{m_1}^{\rho }\right)}+m_1^{V_a\left({\mathbf{Q}}_{m_2}^{\varrho }\right)}$.

$$r-a{\mathfrak{V}}_{\tau }\left(r\right)=m_1-a{\mathfrak{V}}_{\rho }\left(m_1\right)+m_2-a{\mathfrak{V}}_{\varrho }\left(m_2\right).$$

(32)

Relation (32) implies that $r=m_1+m_2$ and then ${\mathfrak{V}}_{\tau }(m_1+m_2)={\mathfrak{V}}_{\rho }\left(m_1\right)+{\mathfrak{V}}_{\varrho }\left(m_2\right)$. According to [20], we obtain

$${\mathfrak{V}}_{\tau }\left(m\right)={\mathfrak{V}}_{\rho }\left(m\right)=\eta m+\varsigma \mathrm{and}{\mathfrak{V}}_{\varrho }\left(m\right)=\eta m\mathrm{for}\eta \ne 0\mathrm{and}\varsigma >0.$$

(33)

The parameter $\varsigma ={\mathfrak{V}}_{\tau }\left(0\right)$ should be strictly positive. In addition, $m_1^{\varrho }\ne 0$. Otherwise, $m\mapsto {\mathfrak{V}}_{\varrho }\left(m\right)=\eta m$ cannot be a VF, as shown in page 6 of [18]. Clearly from (33), measures $\tau$, $\rho$, and $\varrho$ are of the Marchenko–Pastur type law up to affinity. □

4. New Limit Theorems Related to $\begin{array}{|c|}\hline \mathit{a}\\ \hline\end{array}$-Convolution

In this section, we present several new limit theorems concerning $\begin{array}{|c|}\hline a\\ \hline\end{array}$-convolution in connection to free and Boolean additive convolutions. The free additive convolution ⊞ is nothing but the $\begin{array}{|c|}\hline a\\ \hline\end{array}$-convolution with $a=0$. The Boolean additive convolution $\varrho \uplus \rho$ of $\varrho$ and $\rho \in \mathcal{P}$ is defined as the measure given by

$${\mathbb{K}}_{\varrho \uplus \rho }\left(\xi \right)={\mathbb{K}}_{\varrho }\left(\xi \right)+{\mathbb{K}}_{\rho }\left(\xi \right),\xi \in {\mathbb{C}}^{+},$$

where ${\mathbb{K}}_{\rho }\left(\xi \right)=\xi -{\displaystyle \frac{1}{{\mathbb{G}}_{\rho }\left(\xi \right)}}$ represents the Boolean cumulant transform of $\rho$.

$\rho \in \mathcal{P}$ is infinitely divisible with respect to ⊎ if, for every $p\in \mathbb{N}$, there is ${\rho }_p\in \mathcal{P}$ so that

$$\rho =\underset{p\mathrm{times}}{\underbrace{{\rho }_p\uplus \dots \uplus {\rho }_p}}.$$

Every $\rho \in \mathcal{P}$ is infinitely divisible with respect to ⊎, as shown in Theorem 3.6 in [21].

Let $\rho \in {\mathcal{P}}_{ba}$. As elucidated in [16], for $s>0$ so that ${\rho }^{⊞s}$ is defined and ∀ m is sufficiently close to $m_1^{{\rho }^{⊞s}}=s{m}_1^{\sigma }$,

$${\mathfrak{V}}_{{\rho }^{⊞s}}\left(m\right)=s{\mathfrak{V}}_{\rho }(m/s).$$

(34)

In addition, from [22], we know that ∀ $r>0$ and ∀ m sufficiently close to $m_1^{{\rho }^{\uplus r}}=r{m}_1^{\rho }$,

$${\mathfrak{V}}_{{\rho }^{\uplus r}}\left(m\right)=r{\mathfrak{V}}_{\rho }(m/r)+m(m-r{m}_1^{\rho })(1/r-1).$$

(35)

For every $t\ge 0$, the authors in [23] introduced the transformation of measures as follows:

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

They proved that, for $t=1$, the transformation ${\mathbb{B}}_1$ coincides with Bercovici and Pata bijection. We know from [22] that for $\varrho \in {\mathcal{P}}_{ba}$ with finite first moment and ∀ m that is close enough to $m_1^{\varrho }$,

$${\mathfrak{V}}_{{\mathbb{B}}_t\left(\varrho \right)}\left(m\right)={\mathfrak{V}}_{\varrho }\left(m\right)+tm(m-m_1^{\varrho })\mathrm{and}{\mathfrak{V}}_{{\mathbb{B}}_1^{-1}\left(\varrho \right)}\left(m\right)={\mathfrak{V}}_{\varrho }\left(m\right)-m(m-m_1^{\varrho }).$$

(36)

Next, we state and demonstrate the main results of this section.

Theorem 5.

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

(i) 

For $\alpha >0$ so that the measure ${\left({\rho }^{\begin{array}{|c|}\hline a\\ \hline\end{array}{\displaystyle \frac{1}{\alpha }}}\right)}^{⊞\alpha }$ is well defined, we have

$${\left({\rho }^{\begin{array}{|c|}\hline a\\ \hline\end{array}{\displaystyle \frac{1}{\alpha }}}\right)}^{⊞\alpha }\stackrel{\alpha \to +\infty }{\to }{\mathbf{T}}_{a{\mathit{r}}_2\left(\rho \right)}\left(V_a\left(\rho \right)\right),\mathit{in} \mathit{distribution}.$$

(37)

(ii) 

For $\alpha >0$ so that the measure ${\left({\rho }^{⊞{\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\alpha }$ is well defined, we have

$${\left({\rho }^{⊞{\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\alpha }\stackrel{\alpha \to +\infty }{\to }{\mathbf{T}}_{-a{\mathit{r}}_2\left(\rho \right)}\left(V_{-a}\left(\rho \right)\right),\mathit{in} \mathit{distribution}.$$

(38)

(iii) 

For $\alpha >0$ so that the measure ${\left({\rho }^{\begin{array}{|c|}\hline a\\ \hline\end{array}{\displaystyle \frac{1}{\alpha }}}\right)}^{\uplus \alpha }$ is well defined, we have

$${\left({\rho }^{\begin{array}{|c|}\hline a\\ \hline\end{array}{\displaystyle \frac{1}{\alpha }}}\right)}^{\uplus \alpha }\stackrel{\alpha \to +\infty }{\to }{\mathbb{B}}_1^{-1}\left({\mathbf{T}}_{a{\mathit{r}}_2\left(\rho \right)}\left(V_a\left(\rho \right)\right)\right),\mathit{in} \mathit{distribution}.$$

(39)

(iv) 

For $\alpha >0$ so that the measure ${\left({\rho }^{\uplus {\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\alpha }$ is well defined, we have

$${\left({\rho }^{\uplus {\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\alpha }\stackrel{\alpha \to +\infty }{\to }{\mathbb{B}}_1\left({\mathbf{T}}_{-a{\mathit{r}}_2\left(\rho \right)}\left(V_{-a}\left(\rho \right)\right)\right),\mathit{in} \mathit{distribution}.$$

(40)

Proof. 

(i) Using (34) and (15), ∀ m sufficiently close to $m_1^{{\left({\rho }^{\begin{array}{|c|}\hline a\\ \hline\end{array}{\displaystyle \frac{1}{\alpha }}}\right)}^{⊞\alpha }}=m_1^{\rho }$, we have

$$\begin{array}{ccc}\hfill {\mathfrak{V}}_{{\left({\rho }^{\begin{array}{|c|}\hline a\\ \hline\end{array}{\displaystyle \frac{1}{\alpha }}}\right)}^{⊞\alpha }}\left(m\right)& =& \alpha {\mathfrak{V}}_{{\rho }^{\begin{array}{|c|}\hline a\\ \hline\end{array}{\displaystyle \frac{1}{\alpha }}}}(m/\alpha )={\mathfrak{V}}_{\rho }\left(m\right)+a{\mathbf{r}}_2\left(\rho \right)(1-1/\alpha )(m-m_1^{\rho })\hfill \\ & =& \stackrel{\alpha \to +\infty }{\to }{\mathfrak{V}}_{\rho }\left(m\right)+a{\mathbf{r}}_2\left(\rho \right)(m-m_1^{\rho })={\mathfrak{V}}_{{\mathbf{T}}_{a{\mathbf{r}}_2\left(\rho \right)}\left(V_a\left(\rho \right)\right)}\left(m\right).\hfill \end{array}$$

The proof of (37) is concluded by the use of Proposition 4.2 in [15].

(ii) Using (34) and (15), ∀ m sufficiently close to $m_1^{{\left({\rho }^{⊞{\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\alpha }}=m_1^{\rho }$, we have

$$\begin{array}{ccc}\hfill {\mathfrak{V}}_{{\left({\rho }^{⊞{\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\alpha }}\left(m\right)& =& \alpha {\mathfrak{V}}_{{\rho }^{⊞{\displaystyle \frac{1}{\alpha }}}}(m/\alpha )+a{\mathbf{r}}_2\left({\rho }^{⊞{\displaystyle \frac{1}{\alpha }}}\right)(1-\alpha )(m-\alpha m_1^{{\rho }^{⊞{\displaystyle \frac{1}{\alpha }}}})\hfill \\ & =& {\mathfrak{V}}_{\rho }\left(m\right)+{\displaystyle \frac{1}{\alpha }}a{\mathbf{r}}_2\left(\rho \right)(1-\alpha )(m-m_1^{\rho })\hfill \\ & \stackrel{\alpha \to +\infty }{\to }& {\mathfrak{V}}_{\rho }\left(m\right)-a{\mathbf{r}}_2\left(\rho \right)(m-m_1^{\rho })={\mathfrak{V}}_{{\mathbf{T}}_{-a{\mathbf{r}}_2\left(\rho \right)}\left(V_{-a}\left(\rho \right)\right)}\left(m\right).\hfill \end{array}$$

Using Proposition 4.2 in [15], this ends the proof of (38).

(iii) Using (35) and (15), ∀ m sufficiently close to $m_1^{{\left({\rho }^{\begin{array}{|c|}\hline a\\ \hline\end{array}{\displaystyle \frac{1}{\alpha }}}\right)}^{\uplus \alpha }}=m_1^{\rho }$, we have

$$\begin{array}{ccc}\hfill {\mathfrak{V}}_{{\left({\rho }^{\begin{array}{|c|}\hline a\\ \hline\end{array}{\displaystyle \frac{1}{\alpha }}}\right)}^{\uplus \alpha }}\left(m\right)& =& \alpha {\mathfrak{V}}_{{\rho }^{\begin{array}{|c|}\hline a\\ \hline\end{array}{\displaystyle \frac{1}{\alpha }}}}(m/\alpha )+m(m-\alpha m_1^{{\rho }^{\begin{array}{|c|}\hline a\\ \hline\end{array}{\displaystyle \frac{1}{\alpha }}}})(1/\alpha -1)\hfill \\ & =& \alpha \left[{\mathfrak{V}}_{\rho }\left(m\right)/\alpha +a{\mathbf{r}}_2\left(\rho \right)(1-1/\alpha )(m/\alpha -m_1^{\rho }/\alpha )\right]+m(m-m_1^{\rho })(1/\alpha -1)\hfill \\ & =& {\mathfrak{V}}_{\rho }\left(m\right)+a{\mathbf{r}}_2\left(\rho \right)(1-1/\alpha )(m-m_1^{\rho })+m(m-m_1^{\rho })(1/\alpha -1)\hfill \\ & \stackrel{\alpha \to +\infty }{\to }& {\mathfrak{V}}_{\rho }\left(m\right)+a{\mathbf{r}}_2\left(\rho \right)(m-m_1^{\rho })-m(m-m_1^{\rho })={\mathfrak{V}}_{{\mathbb{B}}_1^{-1}\left({\mathbf{T}}_{a{\mathbf{r}}_2\left(\rho \right)}\left(V_a\left(\rho \right)\right)\right)}\left(m\right).\hfill \end{array}$$

Using Proposition 4.2 in [15], the proof of (39) is complete.

(iv) ∀ m sufficiently close to $m_1^{{\left({\rho }^{\uplus {\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\alpha }}=m_1^{\rho }$, Relations (35) and (15) give

$$\begin{array}{ccc}\hfill {\mathfrak{V}}_{{\left({\rho }^{\uplus {\displaystyle \frac{1}{\alpha }}}\right)}^{\begin{array}{|c|}\hline a\\ \hline\end{array}\alpha }}\left(m\right)& =& \alpha {\mathfrak{V}}_{{\rho }^{\uplus {\displaystyle \frac{1}{\alpha }}}}(m/\alpha )+a{\mathbf{r}}_2({\rho }^{\uplus {\displaystyle \frac{1}{\alpha }}})(1-\alpha )(m-\alpha m_1^{{\rho }^{\uplus {\displaystyle \frac{1}{\alpha }}}})\hfill \\ & =& \alpha \left[{\mathfrak{V}}_{\rho }\left(m\right)/\alpha +{\displaystyle \frac{m}{\alpha }}(m/\alpha -m_1^{\rho }/\alpha )(\alpha -1)\right]+a{\displaystyle \frac{1}{\alpha }}{\mathbf{r}}_2\left(\rho \right)(1-\alpha )(m-m_1^{\rho })\hfill \\ & =& {\mathfrak{V}}_{\rho }\left(m\right)+m(m-m_1^{\rho })(1-1/\alpha )+a{\mathbf{r}}_2\left(\rho \right)(1/\alpha -1)(m-m_1^{\rho })\hfill \\ & \stackrel{\alpha \to +\infty }{\to }& {\mathfrak{V}}_{\rho }\left(m\right)+m(m-m_1^{\rho })-a{\mathbf{r}}_2\left(\rho \right)(m-m_1^{\rho })={\mathfrak{V}}_{{\mathbb{B}}_1\left({\mathbf{T}}_{-a{\mathbf{r}}_2\left(\rho \right)}\left(V_{-a}\left(\rho \right)\right)\right)}\left(m\right).\hfill \end{array}$$

This, together with Proposition 4.2 in [15], completes the proof of (40). □

5. ${\mathit{V}}_{\mathit{a}}$-Deformed Free Cumulants and Variance Function

${\left({\mathbf{r}}_n^a\left(\rho \right)\right)}_{n\ge 1}$ denotes the $V_a$-deformed free cumulant of $\rho$ (i.e., the free cumulants of $V_a\left(\rho \right)$). We obtain the following result:

Theorem 6.

Let $\rho \in {\mathcal{P}}_c$. Then, the $V_a$-deformed free cumulant of ρ are ${\mathit{r}}_1^a\left(\rho \right)=m_1^{\rho }-a{\mathit{r}}_2\left(\rho \right)$ and ∀ $n\ge 1$ as follows:

$${\left.{\mathit{r}}_{n+1}^a\left(\rho \right)={\displaystyle \frac{1}{n!}}{\displaystyle \frac{d^{n-1}}{d{t}^{n-1}}}{\left({\mathfrak{V}}_{\rho }(t+a{\mathit{r}}_2\left(\rho \right))+a{\mathit{r}}_2\left(\rho \right)(t+a{\mathit{r}}_2\left(\rho \right)-m_1^{\rho })\right)}^n\right|}_{t=m_1^{\rho }-a{\mathit{r}}_2\left(\rho \right)}.$$

(41)

Proof. 

From Relation (5), it is clear that the $V_a$-deformed free cumulants of $\rho$ are equal to the free cumulants of $V_a\left(\rho \right)$. According to Theorem 3.3 in [15], we have ${\mathbf{r}}_1^a\left(\rho \right)={\mathbf{r}}_1\left(V_a\left(\rho \right)\right)=m_1^{V_a\left(\rho \right)}=m_1^{\rho }-a{\mathbf{r}}_2\left(\rho \right)$ and ∀ $n\ge 1$

$${\left.{\mathbf{r}}_{n+1}^a\left(\rho \right)={\mathbf{r}}_{n+1}\left(V_a\left(\rho \right)\right)={\displaystyle \frac{1}{n!}}{\displaystyle \frac{d^{n-1}}{d{t}^{n-1}}}{\left({\mathfrak{V}}_{V_a\left(\rho \right)}\left(t\right)\right)}^n\right|}_{t=m_1^{V_a\left(\rho \right)}}$$

(42)

Relation (41) follows by combining (42) and (14). □

Corollary 4.

(i) Consider the standard Wigner’s semicircle measure of the form (17) (for $a_1=a_2=0$) with $m_1^{\mu }=0$ and variance ${\mathit{r}}_2\left(\mu \right)=1$. Then, ${\mathit{r}}_1^a\left(\mu \right)=-a$, ${\mathit{r}}_2^a\left(\mu \right)=1$ and ∀ $n\ge 2$, ${\mathit{r}}_{n+1}^a\left(\mu \right)=a^{n-1}$.

(ii) 

Consider the Marchenko–Pastur law of the form (17) (for $a_1\ne 0$ and $a_2=0$) with $m_1^{\mu }=0$ and variance ${\mathit{r}}_2\left(\mu \right)=1$. Then, ${\mathit{r}}_1^a\left(\mu \right)=-a$, ${\mathit{r}}_2^a\left(\mu \right)=1$ and ∀ $n\ge 2$, ${\mathit{r}}_{n+1}^a\left(\mu \right)={(a+a_1)}^{n-1}$.

Corollary 5.

Suppose $\mathfrak{V}(.)$ is analytic near 0 and $\mathfrak{V}\left(0\right)=1$. Then the following conditions are equivalent.

(i) 

A deformed measure $V_a\left(\rho \right)\in {\mathcal{P}}_c$ exists, which is ⊞-infinitely divisible, having mean $-a$ and variance 1 such that $\mathfrak{V}(·)$ is the VF of of the CSK family induced by ρ.

(ii) 

There exists $\omega \in {\mathcal{P}}_c$ such that

$${\left.{\displaystyle \frac{1}{n!}}{\displaystyle \frac{d^{n-1}}{d{t}^{n-1}}}{\left({\mathfrak{V}}_{\rho }(t+a)+a(t+a)\right)}^n\right|}_{t=-a}=\int y^{n-1}\omega \left(dy\right),n\ge 1anda\in \mathbb{R}.$$

(43)

Proof. 

We know that $V_a\left(\rho \right)$ is ⊞-infinitely divisible if and only if there exists $\omega \in {\mathcal{P}}_c$ such that

$${\mathfrak{R}}_{V_a\left(\varrho \right)}\left(z\right)=m_1^{V_a\left(\varrho \right)}+z\int {\displaystyle \frac{\omega \left(dy\right)}{1-zy}}.$$

(44)

Relation (43) follows by combining (41) and (44). □

6. Conclusions

This article provides significant insights into the stability of the free Meixner family of probability measures under the concepts of $V_a$-transformation of a measure and of a convolution. In addition, two novel results are presented by Theorems 3 and 6 that leads to some new properties of the Marchenko–Pastur CSK family. Furthermore, some new limit theorems are proved for the $\begin{array}{|c|}\hline a\\ \hline\end{array}$-convolution basing on the concept of VF and involving both free and Boolean additive convolutions. Also, some properties related to $V_a$-deformed free cumulants are presented in connection with the concept of VF. These studies extend the understanding of $V_a$-transformation in the non-commutative probability framework.