r-Free Convolution and Variance Function

Abstract

The concept of r-free convolution (which is represented by $\begin{array}{|c|}\hline r\\ \hline\end{array}$) was introduced for $0\le r\le 1$. It is equal to the Boolean additive convolution ⊎ if $r=0$ and reduced to the free additive convolution ⊞ when $r=1$. This paper presents certain features of the r-free convolution in relation to the CSK families and their associated variance functions. We provide the variance function formula under $\begin{array}{|c|}\hline r\\ \hline\end{array}$-convolution power. We then estimate members of the r-free Gaussian and r-free Poisson CSK families using the variance function and the $\begin{array}{|c|}\hline r\\ \hline\end{array}$-convolution, respectively. Additionally, a novel limit theorem for the $\begin{array}{|c|}\hline r\\ \hline\end{array}$-convolution is provided utilizing the variance functions and the free multiplicative convolution.

1. Introduction

In the field of operator theory and probability, the study of convolutions plays a pivotal role in understanding the behavior of random variables, stochastic processes, and functional spaces. The concept of free convolution, introduced by Voiculescu in the 1980s, has become a fundamental tool in the study of non-commutative probability theory. Free convolution generalizes the classical notion of convolution by taking into account non-commuting random variables, thus extending the classical framework of independent random variables to the non-commutative setting. However, in many real-world applications, the structure of the problem at hand may not lend itself to the classical or free convolution frameworks. In such cases, more refined models are required to account for deformations in the underlying operations. The deformed free convolution is one such extension, which introduces a parameterized deformation of the free convolution operation. This deformation aims to incorporate additional structural information or constraints, offering a more flexible approach to modeling complex phenomena in both theoretical and applied settings. This topic has been explored and expanded in many ways in a number of articles, see [1,2,3,4,5,6].

A notion of r-free convolution (denoted by $\begin{array}{|c|}\hline r\\ \hline\end{array}$-convolution) was introduced in [7] for $0\le r\le 1$. It is the free additive convolution ⊞ when $r=1$ and equals to the Boolean additive convolution ⊎ if $r=0$. Let ${\mathcal{P}}_c$ be the set of real probabilities with compact support. For $\sigma \in {\mathcal{P}}_c$ and $0\le r\le 1$, let

$$C_r\left(\sigma \right)=r\sigma +(1-r){\delta }_0.$$

Here, ${\delta }_0$ denotes the Dirac mass at 0. Based on the map $C_r$, the r-free convolution $\mu$ of ${\nu }_1$ and ${\nu }_2\in {\mathcal{P}}_c$, that is, $\mu ={\nu }_1\begin{array}{|c|}\hline r\\ \hline\end{array}{\nu }_2$, is defined by (see [8])

$$(\mu ,C_r\left({\nu }_1\right)⊞C_r\left({\nu }_2\right))=({\nu }_1,C_r\left({\nu }_1\right))⊞({\nu }_2,C_r\left({\nu }_2\right)).$$

(1)

The r-deformed free cumulant transform ${\mathcal{R}}_{\nu }^{\left(r\right)}(.)$ is defined by (see [7])

$${\mathcal{R}}_{\nu }^{\left(r\right)}\left({\mathcal{G}}_{C_r\left(\nu \right)}\left(z\right)\right)=z-{\displaystyle \frac{1}{{\mathcal{G}}_{\nu }\left(z\right)}},\forall z\mathrm{in}\mathrm{an}\mathrm{appropriate}\mathrm{domain},$$

(2)

where ${\mathcal{G}}_{\nu }(.)$ is the Cauchy–Stieltjes transformation of $\nu$ given by

$${\mathcal{G}}_{\nu }\left(z\right)=\int {\displaystyle \frac{1}{z-\zeta }}\nu \left(d\zeta \right)z\in \mathbb{C}\setminus \mathrm{supp}\left(\nu \right).$$

(3)

We have

$${\mathcal{R}}_{{\nu }_1\begin{array}{|c|}\hline r\\ \hline\end{array}{\nu }_2}^{\left(r\right)}\left(z\right)={\mathcal{R}}_{{\nu }_1}^{\left(r\right)}\left(z\right)+{\mathcal{R}}_{{\nu }_2}^{\left(r\right)}\left(z\right).$$

The central limiting theorem for $\begin{array}{|c|}\hline r\\ \hline\end{array}$-convolution is proven, resulting in the r-free Gaussian measure. The Poisson limit measure for the $\begin{array}{|c|}\hline r\\ \hline\end{array}$-convolution is also demonstrated and it is called the r-free Poisson measure. Furthermore, an explicit description is given in [4] for the r-free Poisson measure and the corresponding moments. Also, some properties of the moments of the r-free Gaussian measure are provided. Further results related to r-free convolution can be found in [9,10,11,12,13].

This article is devoted to the study of r-free convolution. Our approach is new and is based on a novel concept called the variance function of a Cauchy–Stieltjes Kernel (CSK) family. For the clarity of the results to be presented, we need to recall some facts regarding CSK families: they are about families of probability measures defined similarly to natural exponential families by considering the Cauchy–Stieltjes Kernel ${(1-\theta \zeta )}^{-1}$ replacing $exp\left(\theta \zeta \right)$, the exponential kernel. Some results on CSK families are presented in [14] involving compactly supported measures. Further extensions on CSK families are given in [15,16] to cover measures with one sided support boundary, say from above. Denote by ${\mathcal{P}}_{ba}$ the set of (non-degenerate) probabilities with one sided support boundary from above. Let $\sigma \in {\mathcal{P}}_{ba}$, thus

$${\mathcal{M}}_{\sigma }\left(\theta \right)=\int {\displaystyle \frac{1}{1-\theta \zeta }}\sigma \left(d\zeta \right)$$

(4)

converges ∀$\theta \in [0,{\theta }_{+}^{\sigma })$ with ${\displaystyle \frac{1}{{\theta }_{+}^{\sigma }}}=max\{0,sup\mathrm{supp}\left(\sigma \right)\}$. The CSK family induced by $\sigma$ is the family of measures:

$${\mathcal{K}}_{+}\left(\sigma \right)=\{{\mathbb{P}}_{\theta }^{\sigma }\left(d\zeta \right)={\displaystyle \frac{1}{{\mathcal{M}}_{\sigma }\left(\theta \right)(1-\theta \zeta )}}\sigma \left(d\zeta \right):\theta \in (0,{\theta }_{+}^{\sigma })\}.$$

The function of means $\theta \mapsto k_{\sigma }\left(\theta \right)=\int \zeta {\mathbb{P}}_{\theta }^{\sigma }\left(d\zeta \right)$ is increasing strictly on $(0,{\theta }_{+}^{\sigma })$, see [15]. The mean domain of ${\mathcal{K}}_{+}\left(\sigma \right)$ is the interval $(m_0^{\sigma },m_{+}^{\sigma })=k_{\sigma }\left((0,{\theta }_{+}^{\sigma })\right)$. This gives a re-parametrization of ${\mathcal{K}}_{+}\left(\sigma \right)$ by the mean. Consider ${\psi }_{\sigma }(·)$ the inverse of $k_{\sigma }(·)$. For $m\in (m_0^{\sigma },m_{+}^{\sigma })$, write $Q_m^{\sigma }\left(d\zeta \right)={\mathbb{P}}_{{\psi }_{\sigma }\left(m\right)}^{\sigma }\left(d\zeta \right)$, and we obtain

$${\mathcal{K}}_{+}\left(\sigma \right)=\{Q_m^{\sigma }\left(d\zeta \right);m\in (m_0^{\sigma },m_{+}^{\sigma })\}.$$

Consider

$$B=B\left(\sigma \right)=max\{0,sup\mathrm{supp}\left(\sigma \right)\}={\displaystyle \frac{1}{{\theta }_{+}^{\sigma }}}.$$

(5)

It is proved in [15] that

$$m_0^{\sigma }=\underset{\theta \to 0^{+}}{\lim }{k}_{\sigma }\left(\theta \right)\mathrm{and}{m}_{+}^{\sigma }=B-\underset{z\to B^{+}}{\lim }{\displaystyle \frac{1}{{\mathcal{G}}_{\sigma }\left(z\right)}},$$

(6)

If the support of $\sigma$ is bounded from below, the CSK family will be denoted by ${\mathcal{K}}_{-}\left(\sigma \right)$. We have $\theta \in ({\theta }_{-}^{\sigma },0)$, where ${\theta }_{-}^{\sigma }$ is either $1/A\left(\sigma \right)$ or $-\infty$ with $A=A\left(\sigma \right)=min\{0,infsupp(\sigma \left)\right\}$. The mean domain for ${\mathcal{K}}_{-}\left(\sigma \right)$ is $(m_{-}^{\sigma },m_0^{\sigma })$ with $m_{-}\left(\sigma \right)=A-1/{\mathcal{G}}_{\sigma }\left(A\right)$. If the support of $\sigma$ is compact, then $\theta \in ({\theta }_{-}^{\sigma },{\theta }_{+}^{\sigma })$ and $\mathcal{K}\left(\sigma \right)={\mathcal{K}}_{+}\left(\sigma \right)\cup {\mathcal{K}}_{-}\left(\sigma \right)\cup \left\{\sigma \right\}$ is the two-sided CSK family.

Let $\sigma \in {\mathcal{P}}_{ba}$. The function

$$x\mapsto {\mathcal{V}}_{\sigma }\left(x\right)=\int {(\zeta -x)}^2{Q}_x^{\sigma }\left(d\zeta \right),$$

(7)

is called the variance function (VF) of ${\mathcal{K}}_{+}\left(\sigma \right)$, see [14]. If $m_0^{\sigma }=\int t\sigma \left(dt\right)$ does not exist, all members of ${\mathcal{K}}_{+}\left(\sigma \right)$ have infinite variance. The following substitute is denoted ${\mathbb{V}}_{\sigma }(·)$ and is introduced in [15], as

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

(8)

is said to be the pseudo-variance function (PVF) of ${\mathcal{K}}_{+}\left(\sigma \right)$. If $m_0^{\sigma }$ is finite, then ${\mathcal{V}}_{\sigma }(·)$ exists (see [15]) and

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

(9)

In this article, the concept of $\begin{array}{|c|}\hline r\\ \hline\end{array}$-convolution is investigated in connection with CSK families and the associated VFs. We prove, in Section 2, the expression for the VF under the power of $\begin{array}{|c|}\hline r\\ \hline\end{array}$-convolution. This formula is exploited in Section 3 in approximating members of the r-free Gaussian and the r-free Poisson CSK families. In Section 4, the free multiplicative convolution and the VF are used to demonstrate a new limit theorem for the $\begin{array}{|c|}\hline r\\ \hline\end{array}$-convolution.

We conclude this section by presenting some facts throughout the following remarks to help with the proof of this article’s results.

Remark 1.

(i) 

Let $\phi \left(\sigma \right)$ be the image of σ by $\phi :\zeta ⟼\eta \zeta +\beta$ where $\eta \ne 0$ and $\beta \in \mathbb{R}$. Then, ∀x is close enough to $m_0^{\phi \left(\sigma \right)}=\phi \left(m_0^{\sigma }\right)=\eta m_0^{\sigma }+\beta$:

$${\mathbb{V}}_{\phi \left(\sigma \right)}\left(x\right)={\displaystyle \frac{{\eta }^{2}x}{x-\beta }}{\mathbb{V}}_{\sigma }\left({\displaystyle \frac{x-\beta }{\eta }}\right).$$

(10)

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

$${\mathcal{V}}_{\phi \left(\sigma \right)}\left(x\right)={\eta }^2{\mathcal{V}}_{\sigma }\left({\displaystyle \frac{x-\beta }{\eta }}\right).$$

(11)

(ii) 

From ([15], Corollary 3.6), we have

$$x/{\mathbb{V}}_{\sigma }\left(x\right)\stackrel{x\searrow m_0^{\sigma }}{\to }0and{x}^2/{\mathbb{V}}_{\sigma }\left(x\right)\stackrel{x\searrow m_0^{\sigma }}{\to }0.$$

From ([15], Proposition 3.3), we have

$$x+{\mathbb{V}}_{\sigma }\left(x\right)/x\stackrel{x\searrow m_0^{\sigma }}{\to }\infty andx+{\mathbb{V}}_{\sigma }\left(x\right)/x\stackrel{x\nearrow m_{+}^{\sigma }}{\to }B.$$

2. Variance Function and $\begin{array}{|c|}\hline \mathit{r}\\ \hline\end{array}$-Convolution

Before going to the main result of this section concerning the VF, some needed properties are proved for the ${\mathcal{R}}_{\nu }^{\left(r\right)}$ transform.

Proposition 1.

Let $\nu \in {\mathcal{P}}_c$ be non-degenerate. We have

(i) 

${\mathcal{R}}_{\nu }^{\left(r\right)}$ is increasing strictly on $({\mathcal{G}}_{\nu }\left(A\left(\nu \right)\right),{\mathcal{G}}_{\nu }\left(B\left(\nu \right)\right))$.

(ii) 

For $x\in (m_{-}^{\nu },m_{+}^{\nu })$,

$${\mathcal{R}}_{\nu }^{\left(r\right)}\left({\displaystyle \frac{rx}{{\mathbb{V}}_{\nu }\left(x\right)}}+{\displaystyle \frac{1-r}{x+{\mathbb{V}}_{\nu }\left(x\right)/x}}\right)=x.$$

(12)

(iii) 

${\displaystyle \underset{w\searrow 0}{\lim }{\mathcal{R}}_{\nu }^{\left(r\right)}\left(w\right)=m_0^{\nu }}$.

(iv) 

${\displaystyle \underset{w\searrow 0}{\lim }w{\mathcal{R}}_{\nu }^{\left(r\right)}\left(w\right)=0.}$

Proof. 

(i) Denote $b=supsupp\left(\nu \right)<+\infty$ and $-\infty <a=infsupp\left(\nu \right)$. For all $z>b$, we have

$${\displaystyle \frac{\partial }{\partial w}}{\mathcal{G}}_{C_r\left(\nu \right)}\left(w\right)=-\int {\displaystyle \frac{1}{{(w-y)}^2}}{C}_r\left(\nu \right)\left(dy\right)<0.$$

So that the function $w\mapsto {\mathcal{G}}_{C_r\left(\nu \right)}\left(w\right)$ is strictly decreasing on $(b,+\infty )$.

We see from (2) that

$${\mathcal{R}}_{\nu }^{\left(r\right)}\left({\mathcal{G}}_{C_r\left(\nu \right)}\left(w\right)\right)=w-{\displaystyle \frac{1}{{\mathcal{G}}_{\nu }\left(w\right)}}=K_{\nu }\left(w\right),$$

where $K_{\nu }(·)$ is the Boolean cumulant transform of $\nu$. Furthermore, from ([17], Proposition 2.2(i)), the function $w\mapsto K_{\nu }\left(w\right)$ is strictly decreasing on $(b,+\infty )$. Thus, the function

$$y\mapsto {\mathcal{R}}_{\nu }^{\left(r\right)}\left(y\right):=K_{\nu }\left({\mathcal{G}}_{C_r\left(\nu \right)}^{-1}\left(y\right)\right)$$

is increasing strictly on $(0,{\mathcal{G}}_{\nu }\left(B\left(\nu \right)\right))$ as a composition of two decreasing functions.

Using the same reasoning (by working on $(-\infty ,a))$, we may prove that $y\mapsto {\mathcal{R}}_{\nu }^{\left(r\right)}\left(y\right)$ is increasing strictly on $({\mathcal{G}}_{\nu }\left(A\left(\nu \right)\right),0)$. Then, due to continuity, the function $y\mapsto {\mathcal{R}}_{\nu }^{\left(r\right)}\left(y\right)$ is increasing strictly on $({\mathcal{G}}_{\nu }\left(A\left(\nu \right)\right),{\mathcal{G}}_{\nu }\left(B\left(\nu \right)\right))$.

(ii) For $x\in (m_{-}^{\nu },m_{+}^{\nu })$, we have that

$${\mathcal{G}}_{C_r\left(\nu \right)}\left(x+{\mathbb{V}}_{\nu }\left(x\right)/x\right)=r{\mathcal{G}}_{\nu }\left(x+{\mathbb{V}}_{\nu }\left(x\right)/x\right)+{\displaystyle \frac{1-r}{x+{\mathbb{V}}_{\nu }\left(x\right)/x}}={\displaystyle \frac{rx}{{\mathbb{V}}_{\nu }\left(x\right)}}+{\displaystyle \frac{1-r}{x+{\mathbb{V}}_{\nu }\left(x\right)/x}}.$$

(13)

We see from (2) that

$${\mathcal{R}}_{\nu }^{\left(r\right)}\left({\mathcal{G}}_{C_r\left(\nu \right)}\left(w\right)\right)=w-{\displaystyle \frac{1}{{\mathcal{G}}_{\nu }\left(w\right)}}$$

(14)

Combining (14) with (13), we obtain

$${\mathcal{R}}_{\nu }^{\left(r\right)}\left({\displaystyle \frac{rx}{{\mathbb{V}}_{\nu }\left(x\right)}}+{\displaystyle \frac{1-r}{x+{\mathbb{V}}_{\nu }\left(x\right)/x}}\right)={\mathcal{R}}_{\nu }^{\left(r\right)}\left({\mathcal{G}}_{C_r\left(\nu \right)}\left(x+{\displaystyle \frac{{\mathbb{V}}_{\nu }\left(x\right)}{x}}\right)\right)=x.$$

(iii) From Remark 1, one sees that

$${\displaystyle \underset{w\searrow 0}{\lim }{\mathcal{R}}_{\nu }^{\left(r\right)}\left(w\right)=\underset{x\searrow m_0^{\nu }}{\lim }{\mathcal{R}}_{\nu }^{\left(r\right)}\left(\frac{rx}{{\mathbb{V}}_{\nu }\left(x\right)}+\frac{1-r}{x+{\mathbb{V}}_{\nu }\left(x\right)/x}\right)=m_0^{\nu }.}$$

(iv) Combining Remark 1 with relation (12), we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \underset{w\searrow 0}{\lim }w{\mathcal{R}}_{\nu }^{\left(r\right)}\left(w\right)}& =& {\displaystyle \underset{x\searrow m_0^{\nu }}{\lim }\left(\frac{rx}{{\mathbb{V}}_{\nu }\left(x\right)}+\frac{1-r}{x+{\mathbb{V}}_{\nu }\left(x\right)/x}\right){\mathcal{R}}_{\nu }^{\left(r\right)}\left(\frac{rx}{{\mathbb{V}}_{\nu }\left(x\right)}+\frac{1-r}{x+{\mathbb{V}}_{\nu }\left(x\right)/x}\right)}\hfill \\ & =& {\displaystyle \underset{x\searrow m_0^{\nu }}{\lim }\left(\frac{r{x}^2}{{\mathbb{V}}_{\nu }\left(x\right)}+\frac{1-r}{1+{\mathbb{V}}_{\nu }\left(x\right)/x^2}\right)=0.}\hfill \end{array}$$

□

Next, we write and demonstrate the main result of this section.

Theorem 1.

Let $\nu \in {\mathcal{P}}_c$ be non-degenerate. For $\alpha >0$ so that ${\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }$ is defined, we have

(i) 

${\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }\in {\mathcal{P}}_c$.

(ii) 

∀x is sufficiently close to $m_0^{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}=\alpha m_0^{\nu }$, we have

$${\displaystyle \frac{r{x}^2+{\mathbb{V}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}\left(x\right)}{{\mathbb{V}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}\left(x\right)\left(x+{\mathbb{V}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}\left(x\right)/x\right)}}={\displaystyle \frac{r\frac{x^2}{{\alpha }^2}+{\mathbb{V}}_{\nu }\left(\frac{x}{\alpha }\right)}{{\mathbb{V}}_{\nu }\left(\frac{x}{\alpha }\right)\left(\frac{x}{\alpha }+\alpha {\mathbb{V}}_{\nu }\left(\frac{x}{\alpha }\right)/x\right)}}.$$

(15)

Furthermore,

$${\displaystyle \frac{rx(x-\alpha m_0^{\nu })+{\mathcal{V}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}\left(x\right)}{{\mathcal{V}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}\left(x\right)\left(x(x-\alpha m_0^{\nu })+{\mathcal{V}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}\left(x\right)\right)}}={\displaystyle \frac{\frac{1}{\alpha }\left[\frac{rx}{\alpha }(\frac{x}{\alpha }-m_0^{\nu })+{\mathcal{V}}_{\nu }\left(\frac{x}{\alpha }\right)\right]}{{\mathcal{V}}_{\nu }\left(\frac{x}{\alpha }\right)\left(\frac{x}{\alpha }(\frac{x}{\alpha }-m_0^{\nu })+{\mathcal{V}}_{\nu }\left(\frac{x}{\alpha }\right)\right)}}.$$

(16)

Proof. 

Since $\nu \in {\mathcal{P}}_c$, in a domain that contains some open interval $(-ϵ,ϵ)$, for $ϵ>0$, the transform ${\mathcal{R}}_{\nu }^{\left(r\right)}(.)$ is univalent. Therefore, in the same domain, ${\mathcal{R}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}^{\left(r\right)}(·)=\alpha {\mathcal{R}}_{\nu }^{\left(r\right)}(·)$ is univalent. Thus, ${\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }\in {\mathcal{P}}_c$.

Using Proposition 1(iii),

$${\displaystyle m_0^{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}=\underset{z⟶0}{\lim }{\mathcal{R}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}^{\left(r\right)}\left(z\right)=\underset{z⟶0}{\lim }\alpha {\mathcal{R}}_{\nu }^{\left(r\right)}\left(z\right)=\alpha m_0^{\nu }.}$$

∀x is sufficiently close to $m_0^{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}$ so that $x/\alpha \in (m_{-}^{\nu },m_{+}^{\nu })$ and

$${\displaystyle \frac{rx}{{\mathbb{V}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}\left(x\right)}}+{\displaystyle \frac{1-r}{x+{\mathbb{V}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}\left(x\right)/x}}\in ({\mathcal{G}}_{\nu }\left(A\left(\nu \right)\right),{\mathcal{G}}_{\nu }\left(B\left(\nu \right)\right)),$$

we have

$$\begin{array}{ccc}\hfill {\mathcal{R}}_{\nu }^{\left(r\right)}\left({\displaystyle \frac{rx}{{\mathbb{V}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}\left(x\right)}}+{\displaystyle \frac{1-r}{x+{\mathbb{V}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}\left(x\right)/x}}\right)& =& {\displaystyle \frac{1}{\alpha }}{\mathcal{R}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}^{\left(r\right)}\left({\displaystyle \frac{rx}{{\mathbb{V}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}\left(x\right)}}+{\displaystyle \frac{1-r}{x+{\mathbb{V}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}\left(x\right)/x}}\right)\hfill \\ & =& {\displaystyle \frac{x}{\alpha }}\hfill \\ & =& {\mathcal{R}}_{\nu }^{\left(r\right)}\left({\displaystyle \frac{r\frac{x}{\alpha }}{{\mathbb{V}}_{\nu }\left(\frac{x}{\alpha }\right)}}+{\displaystyle \frac{1-r}{\frac{x}{\alpha }+\alpha {\mathbb{V}}_{\nu }\left(\frac{x}{\alpha }\right)/x}}\right).\hfill \end{array}$$

We have that ${\mathcal{R}}_{\nu }^{\left(r\right)}$ is one-to-one on $({\mathcal{G}}_{\nu }\left(A\left(\nu \right)\right),{\mathcal{G}}_{\nu }\left(B\left(\nu \right)\right))$, see Proposition 1(i). Thus,

$${\displaystyle \frac{rx}{{\mathbb{V}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}\left(x\right)}}+{\displaystyle \frac{1-r}{x+{\mathbb{V}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}\left(x\right)/x}}={\displaystyle \frac{r\frac{x}{\alpha }}{{\mathbb{V}}_{\nu }\left(\frac{x}{\alpha }\right)}}+{\displaystyle \frac{1-r}{\frac{x}{\alpha }+\alpha {\mathbb{V}}_{\nu }\left(\frac{x}{\alpha }\right)/x}},$$

which is nothing but (15). Furthermore, combining (15) and (9), we obtain (16). □

3. $\begin{array}{|c|}\hline \mathit{r}\\ \hline\end{array}$-Convolution and CSK Families Approximations

3.1. The r-Free Gaussian CSK Family Approximation

From ([7], Theorem 5.1), the standard r-free Gaussian law ${\mu }_r$ is

$${\mu }_r\left(d\zeta \right)={\displaystyle \frac{1}{2}}\left(g_r\left(\zeta \right){\mathbf{1}}_{I_r}\left(\zeta \right)+g_r(-\zeta ){\mathbf{1}}_{(-I_r)}\left(\zeta \right)\right),$$

where $g_r\left(\zeta \right)={\displaystyle \frac{1}{\pi \zeta }}\sqrt{4r-{({\zeta }^2-(1+r))}^2}$ and $I_r=[1-\sqrt{r},1+\sqrt{r}]$.

Proposition 2.

∀m is sufficiently close to $m_0^{{\mu }_r}=0$,

$${\mathcal{V}}_{{\mu }_r}\left(m\right)={\displaystyle \frac{1-m^2+\sqrt{{(m^2-1)}^2+4r{m}^2}}{2}}.$$

(17)

Proof. 

According to [7], the limit measure ${\mu }_r$ is such that

$${\mathcal{R}}_{{\mu }_r}^{\left(r\right)}\left(z\right)=z.$$

(18)

Combining (18) and (12), one sees that ∀m is sufficiently close to $m_0^{{\mu }_r}=0$:

$${\displaystyle \frac{rm}{{\mathcal{V}}_{{\mu }_r}\left(m\right)}}+{\displaystyle \frac{1-r}{m+{\mathcal{V}}_{{\mu }_r}\left(m\right)/m}}=m,$$

or equivalently:

$${\mathcal{V}}_{{\mu }_r}^2\left(m\right)+(m^2-1){\mathcal{V}}_{{\mu }_r}\left(m\right)-r{m}^2=0.$$

(19)

Solving Equation (19), knowing that the VF is positive, we obtain the expression given by (17) for ${\mathcal{V}}_{{\mu }_r}(.)$. □

Next, elements of $\mathcal{K}\left({\mu }_r\right)$ are approximated. The dilation of $\nu$ by $e\ne 0$ is denoted as $D_e\left(\nu \right)$.

Theorem 2.

Let $\nu \in {\mathcal{P}}_c$ be non-degenerate with $m_0^{\nu }=0$ and variance one. Then, $\varrho >0$ exists so that if, for $\alpha >0$, the distribution of a random variable $Y_{\alpha }$ is in the CSK family $\mathcal{K}\left({\nu }_{\alpha }\right)$, so that ${\nu }_{\alpha }=D_{1/\alpha }\left({\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }\right)$ and $m/\sqrt{\alpha }$ is the mean of $Y_{\alpha }$ such that $\left|m\right|<\varrho$, then

$$\sqrt{\alpha }{Y}_{\alpha }\stackrel{\alpha \to +\infty }{\to }{Q}_m^{{\mu }_r}\in \mathcal{K}\left({\mu }_r\right)\mathit{in}\mathit{distribution}.$$

Proof. 

Denote by $\mathcal{L}\left(Y_{\alpha }\right)$ the distribution of $Y_{\alpha }$. Since $\mathcal{L}\left(Y_{\alpha }\right)$ is in $\mathcal{K}\left({\nu }_{\alpha }\right)$ with ${\mathcal{V}}_{{\nu }_{\alpha }}\left(m\right)={\mathcal{V}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}\left(\alpha m\right)/{\alpha }^2,$ then $\mathcal{L}\left(\sqrt{\alpha }{Y}_{\alpha }\right)$ belongs to the CSK family with

$${\mathcal{V}}_{\alpha }\left(m\right)=\alpha {\mathcal{V}}_{{\nu }_{\alpha }}(m/\sqrt{\alpha })={\mathcal{V}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}\left(\sqrt{\alpha }m\right)/\alpha .$$

(20)

On the other hand, formula (16) (for $\sqrt{\alpha }m$ instead of m) gives

$${\displaystyle \frac{r\alpha m^2+{\mathcal{V}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}\left(\sqrt{\alpha }m\right)}{{\mathcal{V}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}\left(\sqrt{\alpha }m\right)\left[\alpha m^2+{\mathcal{V}}_{{\nu }^{\begin{array}{|c|}\hline r\\ \hline\end{array}\alpha }}\left(\sqrt{\alpha }m\right)\right]}}={\displaystyle \frac{\frac{1}{\alpha }\left[\frac{r{m}^2}{\alpha }+{\mathcal{V}}_{\nu }(m/\sqrt{\alpha })\right]}{{\mathcal{V}}_{\nu }(m/\sqrt{\alpha })\left[\frac{m^2}{\alpha }+{\mathcal{V}}_{\nu }(m/\sqrt{\alpha })\right]}}.$$

(21)

Using (20), Equation (21) becomes

$${\displaystyle \frac{r{m}^2+{\mathcal{V}}_{\alpha }\left(m\right)}{{\mathcal{V}}_{\alpha }\left(m\right)[m^2+{\mathcal{V}}_{\alpha }\left(m\right)]}}={\displaystyle \frac{\left[\frac{r{m}^2}{\alpha }+{\mathcal{V}}_{\nu }(m/\sqrt{\alpha })\right]}{{\mathcal{V}}_{\nu }(m/\sqrt{\alpha })\left[\frac{m^2}{\alpha }+{\mathcal{V}}_{\nu }(m/\sqrt{\alpha })\right]}}.$$

(22)

Denote by ${\displaystyle v\left(m\right)=\underset{\alpha ⟶+\infty }{\lim }{\mathcal{V}}_{\alpha }\left(m\right)}$. Let $\alpha$ go to $+\infty$ on both sides of Equation (22) (recall that ${\mathcal{V}}_{\nu }\left(0\right)=Var\left(\nu \right)=1$); we obtain $v^2\left(m\right)+(m^2-1)v\left(m\right)-r{m}^2=0$ with nothing but Equation (19). So, with ∀m in a neighborhood of $m_0^{{\mu }_r}=0$, we have ${\displaystyle \underset{\alpha ⟶+\infty }{\lim }{\mathcal{V}}_{\alpha }\left(m\right)={\mathcal{V}}_{{\mu }_r}\left(m\right).}$ The conclusion is made from ([14], Proposition 4.2); that is, $\varrho >0$ exists so that if $\left|m\right|<\varrho$ and $m/\sqrt{\alpha }$ is the mean of $Y_{\alpha }$, then

$$\mathcal{L}\left(\sqrt{\alpha }{Y}_{\alpha }\right)\stackrel{\alpha \to +\infty }{\to }{Q}_m^{{\mu }_r}\in \mathcal{K}\left({\mu }_r\right)\mathrm{in}\mathrm{distribution}.$$

The case $m=0$ is reduced to the central limiting theorem related to $\begin{array}{|c|}\hline r\\ \hline\end{array}$-convolution. □

3.2. Approximation of r-Free Poisson CSK Family

From ([4], Theorem 3.1), the r-free Poisson measure is given, for $\lambda >0$ and $0\le r\le 1$, by

$$p_{\lambda }^{\left(r\right)}\left(dx\right)={\displaystyle \frac{\sqrt{4\lambda r-{(x-(\lambda +1))}^2}}{2\pi rx}}{\mathbf{1}}_{(\lambda +1-2\sqrt{\lambda r},\lambda +1+2\sqrt{\lambda r})}\left(x\right)\left(dx\right)+\left(1-{\displaystyle \frac{1+\lambda -\sqrt{{(\lambda +1)}^2-4\lambda r}}{2r}}\right){\delta }_0.$$

Furthermore, if we consider the sequence of measures

$${\mu }_M=\left(1-{\displaystyle \frac{\lambda }{M}}\right){\delta }_0+{\displaystyle \frac{\lambda }{M}}{\delta }_1,\mathrm{for}M\in {\mathbb{N}}^{*}\mathrm{and}0<\lambda <M.$$

Then

$$\underset{M\mathrm{times}}{\underbrace{{\mu }_M\begin{array}{|c|}\hline r\\ \hline\end{array}{\mu }_M\dots \dots \begin{array}{|c|}\hline r\\ \hline\end{array}{\mu }_M}}\stackrel{M\to +\infty }{\to }{p}_{\lambda }^{\left(r\right)}\mathrm{in}\mathrm{distribution}.$$

Proposition 3.

∀m is sufficiently close to $m_0^{p_{\lambda }^{\left(r\right)}}=\lambda$,

$${\mathcal{V}}_{p_{\lambda }^{\left(r\right)}}\left(m\right)={\displaystyle \frac{1}{2}}\left((\lambda +1)m-m^2+\sqrt{{((\lambda +1)m-m^2)}^2+4r{m}^2(m-\lambda )}\right).$$

(23)

Proof. 

According to [4], we have

$${\mathcal{R}}_{p_{\lambda }^{\left(r\right)}}^{\left(r\right)}\left(z\right)={\displaystyle \frac{\lambda }{1-z}}.$$

(24)

Combining (24) with (12), for m in a neighborhood of $m_0^{p_{\lambda }^{\left(r\right)}}=\lambda$, we obtain

$${\displaystyle \frac{\lambda }{1-\left(\frac{rm}{{\mathbb{V}}_{p_{\lambda }^{\left(r\right)}}\left(m\right)}+\frac{1-r}{m+{\mathbb{V}}_{p_{\lambda }^{\left(r\right)}}\left(m\right)/m}\right)}}=m.$$

(25)

After some calculations, Equation (25) becomes

$$\left({\displaystyle \frac{\lambda }{m}}-1\right){\mathbb{V}}_{p_{\lambda }^{\left(r\right)}}^2\left(m\right)+((\lambda +1)m-m^2){\mathbb{V}}_{p_{\lambda }^{\left(r\right)}}\left(m\right)+r{m}^3=0.$$

(26)

The solutions to Equation (26) are

$${\mathbb{V}}_1\left(m\right)={\displaystyle \frac{m^2-(\lambda +1)m-\sqrt{{((\lambda +1)m-m^2)}^2+4r{m}^2(m-\lambda )}}{2\left(\frac{\lambda }{m}-1\right)}},$$

(27)

and

$${\mathbb{V}}_2\left(m\right)={\displaystyle \frac{m^2-(\lambda +1)m+\sqrt{{((\lambda +1)m-m^2)}^2+4r{m}^2(m-\lambda )}}{2\left(\frac{\lambda }{m}-1\right)}}.$$

(28)

For the expression of the PVF ${\mathbb{V}}_{p_{\lambda }^{\left(r\right)}}(.)$, we have to choose between (27) and (28). It is well known (see ([15], Definition 3.1)) that the ${\mathbb{V}}_{\nu }(.)$ may take negative values. But the VF is always positive. Recall (9) that the function

$${\mathcal{V}}_1\left(m\right)={\displaystyle \frac{m-\lambda }{m}}{\mathbb{V}}_1\left(m\right)={\displaystyle \frac{1}{2}}\left((\lambda +1)m-m^2+\sqrt{{((\lambda +1)m-m^2)}^2+4r{m}^2(m-\lambda )}\right),$$

(29)

is positive in a neighborhood of $\lambda$ and the function

$${\mathcal{V}}_2\left(m\right)={\displaystyle \frac{m-\lambda }{m}}{\mathbb{V}}_2\left(m\right)={\displaystyle \frac{1}{2}}\left((\lambda +1)m-m^2-\sqrt{{((\lambda +1)m-m^2)}^2+4r{m}^2(m-\lambda )}\right),$$

(30)

may take negative values in a neighborhood of $\lambda$. Then, ${\mathcal{V}}_{p_{\lambda }^{\left(r\right)}}(.)$ is provided by (29). □

Next, we approximate elements of $\mathcal{K}\left(p_{\lambda }^{\left(r\right)}\right)$.

Theorem 3.

For $M\in {\mathbb{N}}^{*}$ and $0<\lambda <M$, consider

$${\mu }_M=\left(1-{\displaystyle \frac{\lambda }{M}}\right){\delta }_0+{\displaystyle \frac{\lambda }{M}}{\delta }_1.$$

∀m is sufficiently close to λ,

$$Q_m^{{\mu }_M^{\begin{array}{|c|}\hline r\\ \hline\end{array}M}}\stackrel{M\to +\infty }{\to }{Q}_m^{p_{\lambda }^{\left(r\right)}},indistribution,$$

Proof. 

According to [18], ∀y is sufficiently close to $m_0^{{\mu }_N}=\lambda /N$,

$${\mathcal{V}}_{{\mu }_N}\left(y\right)=y(1-y).$$

(31)

We have that $m_0^{{\mu }_N^{\begin{array}{|c|}\hline r\\ \hline\end{array}N}}=\lambda =m_0^{p_{\lambda }^{\left(r\right)}}$. Then, $\kappa >0$ exists such that

$$\left(m_{-}^{{\mu }_N^{\begin{array}{|c|}\hline r\\ \hline\end{array}N}},m_{+}^{{\mu }_N^{\begin{array}{|c|}\hline r\\ \hline\end{array}N}}\right)\cap \left(m_{-}^{p_{\lambda }^{\left(r\right)}},m_{+}^{p_{\lambda }^{\left(r\right)}}\right)=(\lambda -\kappa ,\lambda +\kappa ).$$

∀$m\in (\lambda -\kappa ,\lambda +\kappa )$

$$\int \xi Q_m^{{\mu }_N^{\begin{array}{|c|}\hline r\\ \hline\end{array}N}}\left(d\xi \right)=m=\int \xi Q_m^{p_{\lambda }^{\left(r\right)}}\left(d\xi \right).$$

Using (16) and (39), we have

$${\displaystyle \frac{rm(m-\lambda )+{\mathcal{V}}_{{\mu }_M^{\begin{array}{|c|}\hline r\\ \hline\end{array}M}}\left(m\right)}{{\mathcal{V}}_{{\mu }_M^{\begin{array}{|c|}\hline r\\ \hline\end{array}M}}\left(m\right)\left(m(m-\lambda )+{\mathcal{V}}_{{\mu }_M^{\begin{array}{|c|}\hline r\\ \hline\end{array}M}}\left(m\right)\right)}}={\displaystyle \frac{rm\left(\frac{m}{M}-\frac{\lambda }{M}\right)+m(1-\frac{m}{M})}{m^2\left(1-\frac{m}{M}\right)\left(1-\frac{\lambda }{M}\right)}}.$$

(32)

Denote by ${\displaystyle {\mathcal{V}}^{\left(r\right)}\left(m\right)=\underset{M⟶+\infty }{\lim }{\mathcal{V}}_{{\mu }_M^{\begin{array}{|c|}\hline r\\ \hline\end{array}M}}\left(m\right)}$. Let M go to $+\infty$ on both sides of (32) and one obtains

$${\displaystyle \frac{rm(m-\lambda )+{\mathcal{V}}^{\left(r\right)}\left(m\right)}{{\mathcal{V}}^{\left(r\right)}\left(m\right)\left(m(m-\lambda )+{\mathcal{V}}^{\left(r\right)}\left(m\right)\right)}}={\displaystyle \frac{1}{m}}.$$

(33)

That is,

$${\left({\mathcal{V}}^{\left(r\right)}\left(m\right)\right)}^2+(m^2-(\lambda +1)m){\mathcal{V}}^{\left(r\right)}\left(m\right)+r{m}^2(\lambda -m)=0.$$

(34)

The solutions of (34) are

$${\mathcal{V}}_1^{\left(r\right)}\left(m\right)={\displaystyle \frac{1}{2}}\left((\lambda +1)m-m^2+\sqrt{{(m^2-(\lambda +1)m)}^2+4r{m}^2(m-\lambda )}\right),$$

(35)

and

$${\mathcal{V}}_2^{\left(r\right)}\left(m\right)={\displaystyle \frac{1}{2}}\left((\lambda +1)m-m^2-\sqrt{{(m^2-(\lambda +1)m)}^2+4r{m}^2(m-\lambda )}\right).$$

(36)

As the function ${\mathcal{V}}^{\left(r\right)}(.)$ is the limit of a sequence of VFs, then it is a VF (see ([14], Proposition 4.2)) and it must be positive in a neighborhood of $\lambda$. Then,

$${\mathcal{V}}^{\left(r\right)}\left(m\right)={\mathcal{V}}_1^{\left(r\right)}\left(m\right)={\displaystyle \frac{1}{2}}\left((\lambda +1)m-m^2+\sqrt{{(m^2-(\lambda +1)m)}^2+4r{m}^2(m-\lambda )}\right),$$

(37)

which is nothing but the expression of the VF, given by (23). Thus,

$${\displaystyle \underset{N⟶+\infty }{\lim }{\mathcal{V}}_{{\mu }_N^{\begin{array}{|c|}\hline r\\ \hline\end{array}N}}\left(m\right)={\mathcal{V}}^{\left(r\right)}\left(m\right)={\mathcal{V}}_{p_{\lambda }^{\left(r\right)}}\left(m\right).}$$

This together with ([14], Proposition 4.2) applied to $Q_m^{{\mu }_N^{\begin{array}{|c|}\hline r\\ \hline\end{array}N}}$ implies that

$$Q_m^{{\mu }_M^{\begin{array}{|c|}\hline r\\ \hline\end{array}M}}\stackrel{M\to +\infty }{\to }{Q}_m^{p_{\lambda }^{\left(r\right)}}\mathrm{in}\mathrm{distribution},\forall m\in (\lambda -\kappa ,\lambda +\kappa ).$$

The case $m=\lambda$ is reduced to the Poisson limit theorem associated with the $\begin{array}{|c|}\hline r\\ \hline\end{array}$-convolution. □

4. A Limit Theorem with Respect to $\begin{array}{|c|}\hline \mathit{r}\\ \hline\end{array}$-Convolution

In this section, by means of the free multiplicative convolution together with the VFs, a new limiting theorem is showed for the $\begin{array}{|c|}\hline r\\ \hline\end{array}$-convolution. ${\mathcal{P}}^{+}$ will denote the set of measures supported on ${\mathbb{R}}_{+}$. ${\mathcal{P}}_c^{+}$ will denote the subset of compactly supported measures from ${\mathcal{P}}^{+}$. For a non-degenerate measure $\rho \in {\mathcal{P}}^{+}$, the definition of the $\mathcal{S}$-transform is provided by

$${\mathcal{M}}_{\rho }\left({\displaystyle \frac{\zeta }{1+\zeta }}{\mathcal{S}}_{\rho }\left(\zeta \right)\right)=1+\zeta ,\forall \zeta \mathrm{sufficiently}\mathrm{close}\mathrm{to}0.$$

For ${\rho }_1$, ${\rho }_2\in {\mathcal{P}}^{+}$, the free multiplicative convolution ${\rho }_1⊠{\rho }_2$ is introduced by ${\mathcal{S}}_{{\rho }_1⊠{\rho }_2}\left(\zeta \right)={\mathcal{S}}_{{\rho }_1}\left(\zeta \right){\mathcal{S}}_{{\rho }_2}\left(\zeta \right)$. Powers ${\rho }^{⊠r}$ of free multiplicative convolution are well defined at least ∀$r\ge 1$ by ${\mathcal{S}}_{{\rho }^{⊠r}}\left(\zeta \right)={\mathcal{S}}_{\rho }{\left(\zeta \right)}^r$, see ([19], Theorem 2.17).

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

Theorem 4.

Let $\nu \in {\mathcal{P}}_c^{+}$ be non-degenerate. Denoting $\gamma ={\displaystyle \frac{Var\left(\nu \right)}{{\left(m_0^{\nu }\right)}^2}}={\displaystyle \frac{{\mathcal{V}}_{\nu }\left(m_0^{\nu }\right)}{{\left(m_0^{\nu }\right)}^2}}$, then

$$D_{1/\left(q{\left(m_0^{\nu }\right)}^q\right)}{\left({\nu }^{⊠q}\right)}^{\begin{array}{|c|}\hline r\\ \hline\end{array}q}\stackrel{q\to +\infty }{\to }{\tau }_{\gamma }indistribution,$$

where ${\tau }_{\gamma }$ is such that

$${\mathcal{V}}_{{\tau }_{\gamma }}\left(m\right)={\displaystyle \frac{1-\frac{ln\left(m\right)}{\gamma }+\sqrt{{\left(\frac{ln\left(m\right)}{\gamma }-1\right)}^2+\frac{4rln\left(m\right)}{\gamma }}}{\frac{2ln\left(m\right)}{\gamma m(m-1)}}},\forall missufficientlycloseto{m}_0^{{\tau }_{\gamma }}=1.$$

(38)

Proof. 

Based on Theorem 1 and ([20], Theorem 2.4 (i)), we have

$$m_0^{D_{1/\left(q{\left(m_0^{\nu }\right)}^q\right)}{\left({\nu }^{⊠q}\right)}^{\begin{array}{|c|}\hline r\\ \hline\end{array}q}}={\displaystyle \frac{m_0^{\left({\left({\nu }^{⊠q}\right)}^{\begin{array}{|c|}\hline r\\ \hline\end{array}q}\right)}}{\left(q{\left(m_0^{\nu }\right)}^q\right)}}={\displaystyle \frac{q{m}_0^{\left({\nu }^{⊠q}\right)}}{q{\left(m_0^{\nu }\right)}^q}}=1.$$

Furthermore, using (11), one can see that ∀m in a small neighborhood of one,

$${\mathcal{V}}_{D_{1/\left(q{\left(m_0^{\nu }\right)}^q\right)}{\left({\nu }^{⊠q}\right)}^{\begin{array}{|c|}\hline r\\ \hline\end{array}q}}\left(m\right)={\displaystyle \frac{1}{q^2{\left(m_0^{\nu }\right)}^{2q}}}{\mathcal{V}}_{{\left({\nu }^{⊠q}\right)}^{\begin{array}{|c|}\hline r\\ \hline\end{array}q}}\left(qm{\left(m_0^{\nu }\right)}^q\right).$$

(39)

From (16), we have that

$$\begin{array}{c}{\displaystyle \frac{rqm{\left(m_0^{\nu }\right)}^q(qm{\left(m_0^{\nu }\right)}^q-q{m}_0^{{\nu }^{⊠q}})+{\mathcal{V}}_{{\left({\nu }^{⊠q}\right)}^{\begin{array}{|c|}\hline r\\ \hline\end{array}q}}\left(qm{\left(m_0^{\nu }\right)}^q\right)}{{\mathcal{V}}_{{\left({\nu }^{⊠q}\right)}^{\begin{array}{|c|}\hline r\\ \hline\end{array}q}}\left(qm{\left(m_0^{\nu }\right)}^q\right)\left(qm{\left(m_0^{\nu }\right)}^q(qm{\left(m_0^{\nu }\right)}^q-q{m}_0^{{\nu }^{⊠q}})+{\mathcal{V}}_{{\left({\nu }^{⊠q}\right)}^{\begin{array}{|c|}\hline r\\ \hline\end{array}q}}\left(qm{\left(m_0^{\nu }\right)}^q\right)\right)}}=\hfill \\ \hfill {\displaystyle \frac{\frac{1}{q}\left[rm{\left(m_0^{\nu }\right)}^q(m{\left(m_0^{\nu }\right)}^q-m_0^{{\nu }^{⊠q}})+{\mathcal{V}}_{{\nu }^{⊠q}}\left(m{\left(m_0^{\nu }\right)}^q\right)\right]}{{\mathcal{V}}_{{\nu }^{⊠q}}\left(m{\left(m_0^{\nu }\right)}^q\right)\left[m{\left(m_0^{\nu }\right)}^q(m{\left(m_0^{\nu }\right)}^q-m_0^{{\nu }^{⊠q}})+{\mathcal{V}}_{{\nu }^{⊠q}}\left(m{\left(m_0^{\nu }\right)}^q\right)\right]}}.\end{array}$$

(40)

That is,

$$\begin{array}{c}{\displaystyle \frac{rm(m-1)+\frac{1}{q^2{\left(m_0^{\nu }\right)}^{2q}}{\mathcal{V}}_{{\left({\nu }^{⊠q}\right)}^{\begin{array}{|c|}\hline r\\ \hline\end{array}q}}\left(qm{\left(m_0^{\nu }\right)}^q\right)}{\frac{1}{q^2{\left(m_0^{\nu }\right)}^{2q}}{\mathcal{V}}_{{\left({\nu }^{⊠q}\right)}^{\begin{array}{|c|}\hline r\\ \hline\end{array}q}}\left(qm{\left(m_0^{\nu }\right)}^q\right)\left(m(m-1)+\frac{1}{q^2{\left(m_0^{\nu }\right)}^{2q}}{\mathcal{V}}_{{\left({\nu }^{⊠q}\right)}^{\begin{array}{|c|}\hline r\\ \hline\end{array}q}}\left(qm{\left(m_0^{\nu }\right)}^q\right)\right)}}=\hfill \\ \hfill {\displaystyle \frac{q{\left(m_0^{\nu }\right)}^{2q}\left[rm{\left(m_0^{\nu }\right)}^{2q}(m-1)+{\mathcal{V}}_{{\nu }^{⊠q}}\left(m{\left(m_0^{\nu }\right)}^q\right)\right]}{{\mathcal{V}}_{{\nu }^{⊠q}}\left(m{\left(m_0^{\nu }\right)}^q\right)\left[m{\left(m_0^{\nu }\right)}^{2q}(m-1)+{\mathcal{V}}_{{\nu }^{⊠q}}\left(m{\left(m_0^{\nu }\right)}^q\right)\right]}}.\end{array}$$

(41)

We come now to the calculation of the limit, when q goes to $+\infty$, of the right side part of (41). By the use of ([20], Theorem 2.4 (ii)), we obtain

$$\begin{array}{c}{\displaystyle \frac{q{\left(m_0^{\nu }\right)}^{2q}\left[rm{\left(m_0^{\nu }\right)}^{2q}(m-1)+{\mathcal{V}}_{{\nu }^{⊠q}}\left(m{\left(m_0^{\nu }\right)}^q\right)\right]}{{\mathcal{V}}_{{\nu }^{⊠q}}\left(m{\left(m_0^{\nu }\right)}^q\right)\left[m{\left(m_0^{\nu }\right)}^{2q}(m-1)+{\mathcal{V}}_{{\nu }^{⊠q}}\left(m{\left(m_0^{\nu }\right)}^q\right)\right]}}\hfill \\ ={\displaystyle \frac{q{\left(m_0^{\nu }\right)}^{2q}\left[rm{\left(m_0^{\nu }\right)}^{2q}(m-1)+\frac{{\left(m_0^{\nu }\right)}^{2q}(m-1)m^{1-1/q}}{(m^{1/q}-1)}\frac{{\mathcal{V}}_{\nu }\left(m^{1/q}\left(m_0^{\nu }\right)\right)}{{\left(m_0^{\nu }\right)}^2}\right]}{\frac{{\left(m_0^{\nu }\right)}^{2q}(m-1)m^{1-1/q}}{(m^{1/q}-1)}\frac{{\mathcal{V}}_{\nu }\left(m^{1/q}\left(m_0^{\nu }\right)\right)}{{\left(m_0^{\nu }\right)}^2}\left[m{m}_0^{2q}(m-1)+\frac{m_0^{2q}(m-1)m^{1-1/q}}{(m^{1/q}-1)}\frac{{\mathcal{V}}_{\nu }\left(m^{1/q}{m}_0\right)}{m_0^2}\right]}}\\ \hfill ={\displaystyle \frac{\left[rm(m-1)(m^{1/q}-1)+(m-1)m^{1-1/q}\frac{{\mathcal{V}}_{\nu }\left(m^{1/q}\left(m_0^{\nu }\right)\right)}{{\left(m_0^{\nu }\right)}^2}\right]}{(m-1)m^{1-1/q}\frac{{\mathcal{V}}_{\nu }\left(m^{1/q}\left(m_0^{\nu }\right)\right)}{{\left(m_0^{\nu }\right)}^2}\left[\frac{m(m-1)}{q}+\frac{(m-1)m^{1-1/q}}{\frac{(m^{1/q}-1)}{1/q}}\frac{{\mathcal{V}}_{\nu }\left(m^{1/q}\left(m_0^{\nu }\right)\right)}{{\left(m_0^{\nu }\right)}^2}\right]}}\stackrel{q\to +\infty }{\to }{\displaystyle \frac{ln\left(m\right)}{\gamma m(m-1)}}.\end{array}$$

(42)

Denote by

$${\displaystyle T\left(m\right)=\underset{q⟶+\infty }{\lim }{\mathcal{V}}_{D_{1/\left(q{\left(m_0^{\nu }\right)}^q\right)}{\left({\nu }^{⊠q}\right)}^{\begin{array}{|c|}\hline r\\ \hline\end{array}q}}\left(m\right)=\underset{q⟶+\infty }{\lim }\frac{1}{q^2{\left(m_0^{\nu }\right)}^{2q}}{\mathcal{V}}_{{\left({\nu }^{⊠q}\right)}^{\begin{array}{|c|}\hline r\\ \hline\end{array}q}}\left(qm{\left(m_0^{\nu }\right)}^q\right).}$$

When q goes to $+\infty$, the left side part of (41) is

$${\displaystyle \frac{rm(m-1)+T\left(m\right)}{T\left(m\right)\left[m\right(m-1)+T(m\left)\right]}}.$$

(43)

Combining (42) with (43), ∀m is sufficiently close to one, and one obtains

$$T\left(m\right)={\displaystyle \frac{1-\frac{ln\left(m\right)}{\gamma }+\sqrt{{\left(\frac{ln\left(m\right)}{\gamma }-1\right)}^2+\frac{4rln\left(m\right)}{\gamma }}}{\frac{2ln\left(m\right)}{\gamma m(m-1)}}}.$$

(44)

According to ([14], Proposition 4.2), since $T(.)$ is a limit of a sequence of VFs, it is a VF that corresponds to a probability measure, which we denote by ${\tau }_{\gamma }$. Then, we have

$$D_{1/\left(q{\left(m_0^{\nu }\right)}^q\right)}{\left({\nu }^{⊠q}\right)}^{\begin{array}{|c|}\hline r\\ \hline\end{array}q}\stackrel{q\to +\infty }{\to }{\tau }_{\gamma }\mathrm{in}\mathrm{distribution},$$

where

$${\mathcal{V}}_{{\tau }_{\gamma }}\left(m\right)={\displaystyle \frac{1-\frac{ln\left(m\right)}{\gamma }+\sqrt{{\left(\frac{ln\left(m\right)}{\gamma }-1\right)}^2+\frac{4rln\left(m\right)}{\gamma }}}{\frac{2ln\left(m\right)}{\gamma m(m-1)}}},\mathrm{and}{m}_0^{{\tau }_{\gamma }}=m_0^{D_{1/\left(q{\left(m_0^{\nu }\right)}^q\right)}{\left({\nu }^{⊠q}\right)}^{\begin{array}{|c|}\hline r\\ \hline\end{array}q}}=1.$$

□

5. Conclusions

A powerful approach to understanding the r-free convolution is through the CSK families of probability measures. These families allow for an algebraic formulation of convolutions in terms of analytic functions. Additionally, the VF provides a useful perspective by characterizing the fluctuations of associated probability measures and revealing structural properties of the convolution operation. In this article, we presented the formula for the VF while considering the power of $\begin{array}{|c|}\hline r\\ \hline\end{array}$-convolution. This formula is used to approximate members of the r-free Gaussian and r-free Poisson CSK families, respectively. In addition, a novel limit theorem for the $\begin{array}{|c|}\hline r\\ \hline\end{array}$-convolution is presented using free multiplicative convolution and VFs. These studies advance our understanding of $\begin{array}{|c|}\hline r\\ \hline\end{array}$-convolution in non-commutative probability.

The study of r-free convolution within the CSK families of probability measures may provide insightful information for practical applications. It may be useful in a number of fields, such as machine learning, dynamical models [21], signal processing, and finance, where it makes data aggregation, a spectrum analysis of complex systems, and the modeling of associated uncertainty easier. In complicated stochastic contexts, r-free convolution is a potent tool for improving mathematical modeling and decision making by offering a non-commutative framework for probability distributions.