Some Limiting Measures Related to Free and Boolean Convolutions

Abstract

Several authors are interested in the study of limiting distributions for large symmetrical random matrices. Our approach of studying limiting distributions is different and is related to the new concept of variance functions of Cauchy–Stieltjes Kernel (CSK) families of probabilities. By means of the variance functions machinery, some novel limiting probability measures are provided involving the free multiplicative law of large numbers, using both free and Boolean (additive and multiplicative) convolutions. Several examples of these limiting probability measures are given in the context of free probability.

1. Introduction

The topic of limiting spectral distribution for symmetric matrices has received much attention from researchers, see [1,2,3,4,5]. Connected to this topic, several works have explored limiting measures for Boolean and free (additive and multiplicative) convolutions during the past few decades; see [6,7,8,9,10]. Many fundamental characteristics for sums and products of classical independent random variables apply symmetrically to free and Boolean independent random variables. In [11], the author presented research on the distribution of sums and products of Boolean independent random variables in an infinitesimal triangular array. The limit measures of Boolean convolutions are dictated by those of free convolutions, and vice versa. These findings demonstrate symmetries and similarities between the limiting distributional behavior of Boolean and classical convolutions. The authors of [12] presented the distributional model behavior of sums of identically distributed free random variables. The limiting measures of classical and free additive convolutions are explicitly related. Tucci [13] established the free multiplicative law of large numbers (FMLLN) for probabilities with bounded support. The authors of [14] presented an expanded result of [13] to cover measures with unbounded support. Further results on the topic of limiting laws in free probability are presented in [15,16,17]. Pursuing the study of this topic, and based on a new concept called variance functions of Cauchy-Stieltjes Kernel (CSK) families, in this paper, we provide some new limiting measures in connection with the FMLLN and involving Boolean and free (additive and multiplicative) convolutions. To illustrate our findings clearly, we need to review key ideas in free probability. $\mathcal{P}$ (respectively, ${\mathcal{P}}_{+}$) represents the set of probabilities on $\mathbb{R}$ (respectively, ${\mathbb{R}}_{+}$). The Cauchy–Stieltjes transform ${\mathbf{G}}_{\sigma }(·)$ of $\sigma \in \mathcal{P}$ is introduced as

$${\mathbf{G}}_{\sigma }\left(w\right)=\int {\displaystyle \frac{1}{w-\xi }}\sigma \left(d\xi \right),\mathrm{for}w\in \mathbb{C}\setminus \mathrm{supp}\left(\sigma \right).$$

The free additive convolution $\sigma ⊞\rho$ of $\sigma$ and $\rho$$\in \mathcal{P}$, is introduced by

$${\mathcal{C}}_{\sigma ⊞\rho }\left(\xi \right)={\mathcal{C}}_{\sigma }\left(\xi \right)+{\mathcal{C}}_{\rho }\left(\xi \right),$$

where ${\mathcal{C}}_{\sigma }(·)$ is the free cumulant transform of $\sigma$ provided by (see [18])

$${\mathcal{C}}_{\sigma }\left({\mathbf{G}}_{\sigma }\left(\xi \right)\right)=\xi -{\displaystyle \frac{1}{{\mathbf{G}}_{\sigma }\left(\xi \right)}},\forall \xi \mathrm{in} \mathrm{an} \mathrm{appropriate} \mathrm{domain}.$$

$\sigma \in \mathcal{P}$ is ⊞-infinitely divisible if for each $p\in \mathbb{N}$, ${\sigma }_p\in \mathcal{P}$ exists, so that

$$\sigma =\underset{p\mathrm{times}}{\underbrace{{\sigma }_p⊞\dots \dots ⊞{\sigma }_p}}.$$

The s-fold additive free convolution of $\sigma$ with itself is denoted by ${\sigma }^{⊞s}$. It is well defined ∀ $s\ge 1$, (see [19]) and

$${\mathcal{C}}_{{\sigma }^{⊞s}}\left(\xi \right)=s{\mathcal{C}}_{\sigma }\left(\xi \right).$$

$\sigma \in \mathcal{P}$ is ⊞-infinitely divisible if ${\sigma }^{⊞s}$ is well defined ∀$s>0$.

For $\lambda \in {\mathcal{P}}_{+}$, $(\lambda \ne {\delta }_0)$, the $\mathbf{S}$-transform is provided by

$${\mathcal{C}}_{\lambda }\left(\zeta {\mathbf{S}}_{\lambda }\left(\zeta \right)\right)={\displaystyle \frac{1}{{\mathbf{S}}_{\lambda }\left(\zeta \right)}}\forall \zeta \mathrm{close} \mathrm{enough} \mathrm{to}0.$$

Multiplying $\mathbf{S}$-transforms results in another $\mathbf{S}$-transform. The multiplicative free convolution $\lambda ⊠\tau$ of $\lambda$ and $\tau$$\in {\mathcal{P}}_{+}$ is introduced by ${\mathbf{S}}_{\lambda ⊠\tau }\left(\zeta \right)={\mathbf{S}}_{\lambda }\left(\zeta \right){\mathbf{S}}_{\tau }\left(\zeta \right)$. Powers of multiplicative free convolution ${\lambda }^{⊠q}$ are well defined at least ∀$q\ge 1$ (see [20], Theorem 2.17), by ${\mathbf{S}}_{{\lambda }^{⊠q}}\left(\zeta \right)={\mathbf{S}}_{\lambda }{\left(\zeta \right)}^q$.

The Boolean additive convolution $\sigma \uplus \rho$ of $\sigma$ and $\rho \in \mathcal{P}$ is the measure provided by

$${\mathbf{E}}_{\sigma \uplus \rho }\left(\xi \right)={\mathbf{E}}_{\sigma }\left(\xi \right)+{\mathbf{E}}_{\rho }\left(\xi \right),\xi \in {\mathbb{C}}^{+},$$

where ${\mathbf{E}}_{\sigma }\left(\xi \right)=\xi -{\displaystyle \frac{1}{{\mathbf{G}}_{\sigma }\left(\xi \right)}},$ is the Boolean cumulant transform of $\sigma$.

$\sigma \in \mathcal{P}$ is ⊎-infinitely divisible if for every $p\in \mathbb{N}$, ${\sigma }_p\in \mathcal{P}$ exists, so that

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

All measures $\sigma \in \mathcal{P}$ are ⊎-infinitely divisible; see ([21], Theorem 3.6).

A multiplicative version of the additive Boolean convolution was defined in [22]. For $\lambda \in {\mathcal{P}}_{+}$, the function

$${\mathsf{\Psi }}_{\lambda }\left(w\right)={\int }_0^{+\infty }{\displaystyle \frac{wy}{1-wy}}\lambda \left(dy\right),w\in \mathbb{C}\setminus {\mathbb{R}}_{+}$$

is univalent in $i{\mathbb{C}}_{+}$. We know that ${\mathsf{\Psi }}_{\lambda }\left(i{\mathbb{C}}_{+}\right)$ is contained in the circle with diameter $\left(\lambda \right(\left\{0\right\})-1,0)$. Furthermore, $\mathbb{R}\cap {\mathsf{\Psi }}_{\lambda }\left(i{\mathbb{C}}_{+}\right)=(\lambda \left(\left\{0\right\}\right)-1,0)$. We have that ${\mathsf{\Psi }}_{\lambda }\left(w^{-1}\right)=w{\mathbf{G}}_{\lambda }\left(w\right)-1.$ According to [23], the $\eta$-transform of $\lambda \in {\mathcal{P}}_{+}$ is introduced as:

$${\eta }_{\lambda }:\mathbb{C}\setminus {\mathbb{R}}_{+}⟶\mathbb{C}\setminus {\mathbb{R}}_{+};w\mapsto {\eta }_{\lambda }\left(w\right)={\displaystyle \frac{{\mathsf{\Psi }}_{\lambda }\left(w\right)}{1+{\mathsf{\Psi }}_{\lambda }\left(w\right)}}.$$

We known that ${\eta }_{\lambda }\left((-\infty ,0)\right)\subset (-\infty ,0)$, ${\displaystyle \underset{w\to 0,w<0}{lim}{\eta }_{\lambda }\left(w\right)={\eta }_{\lambda }\left(0^{-}\right)=0}$ and ${\eta }_{\lambda }\left(\overline{w}\right)=\overline{{\eta }_{\lambda }\left(w\right)}$, for $w\in \mathbb{C}\setminus {\mathbb{R}}_{+}$. Furthermore, $arg\left(w\right)\le arg\left({\eta }_{\lambda }\left(w\right)\right)<\pi$, for $w\in {\mathbb{C}}^{+}$.

The transform

$${\mathbf{B}}_{\lambda }\left(w\right)={\displaystyle \frac{w}{{\eta }_{\lambda }\left(w\right)}}$$

is well defined for $w\in \mathbb{C}\setminus {\mathbb{R}}_{+}$. The Boolean multiplicative convolution of $\lambda$ and $\tau \in {\mathcal{P}}_{+}$ is the unique measure in ${\mathcal{P}}_{+}$, denoted by $\lambda \cup \times \tau$, which satisfies

$${\mathbf{B}}_{\lambda \cup \times \tau }\left(w\right)={\mathbf{B}}_{\lambda }\left(w\right){\mathbf{B}}_{\tau }\left(w\right),\mathrm{for}w\in \mathbb{C}\setminus {\mathbb{R}}_{+}.$$

Note that for $\lambda ,\tau \in {\mathcal{P}}_{+}$, which satisfies

(i)

$arg\left({\eta }_{\lambda }\left(w\right)\right)+arg\left({\eta }_{\tau }\left(w\right)\right)-arg\left(w\right)<\pi \mathrm{for}w\in {\mathbb{C}}^{+}\cup (-\infty ,0)$

(ii)

at least one of the first moments of one of measures $\lambda$ or $\tau$ is finite,

So, $\lambda \cup \times \tau \in {\mathcal{P}}_{+}$ is well defined. It was proven in [22] that ${\mathcal{P}}_{+}$ is not preserved by the multiplicative Boolean convolution. However, the multiplicative Boolean convolution power ${\lambda }^{\cup \times t}\in {\mathcal{P}}_{+}$ still exists for $0\le t\le 1$.

Now, we present the purpose of this article. In free probability, FMLLN is investigated and formulated in [14] for positive free random variables. By means of probabilities, it may be seen as the convergence of

$${\left({\lambda }^{⊠q}\right)}^{{\displaystyle \frac{1}{q}}},q\in \mathbb{N}$$

(1)

where $\lambda \in {\mathcal{P}}_{+}$ and ${\mu }^{\alpha }$ are the push-forward of measure $\mu$ by $y\mapsto y^{\alpha }$ for $\alpha \in \mathbb{R}$. For the sequence (1), the limit law exists; it is denoted by $\mathsf{\Phi }\left[\lambda \right]$. If $\lambda \ne {\delta }_0$, the limit law is determined by

$$\mathsf{\Phi }\left[\lambda \right]\left(\left[0,{\displaystyle \frac{1}{{\mathbf{S}}_{\lambda }(y-1)}}\right]\right)=y,y\in (\lambda \left(\left\{0\right\}\right),1)\mathrm{and}\mathsf{\Phi }\left[\lambda \right]\left(\left\{0\right\}\right)=\lambda \left(\left\{0\right\}\right).$$

In ([24], Theorem 3.1), a description is presented for the FMLLN $\mathsf{\Phi }\left[\lambda \right]$ based on the pseudo-variance function ${\mathbb{V}}_{\lambda }(·)$ of the CSK family induced by $\lambda$; see the next paragraph for these concepts. Some specific examples are provided for $\mathsf{\Phi }\left[\lambda \right]$; see [24].

In this article, we aim to expressly provide, by means of ${\mathbb{V}}_{\lambda }(·)$, the limiting measures $\mathsf{\Phi }\left[{\lambda }^{\cup \times \alpha }\right]$, $\mathsf{\Phi }\left[{\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)}^{\cup \times \alpha }\right]$, $\mathsf{\Phi }\left[{\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)}^{\cup \times \alpha }\right]$, $\mathsf{\Phi }\left[{\left({\mathbb{B}}_r\left(\lambda \right)\right)}^{\cup \times \alpha }\right]$, $\mathsf{\Phi }\left[{\mathbb{M}}_t\left(\lambda \right)\right]$, $\mathsf{\Phi }\left[{\mathbb{M}}_t\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)\right]$, $\mathsf{\Phi }\left[{\mathbb{M}}_t\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)\right]$ and $\mathsf{\Phi }\left[{\mathbb{M}}_t\left({\mathbb{B}}_r\left(\lambda \right)\right)\right]$ for $t>0$, $r>0$, $s>1$ and $\alpha \in [0,1]$. Here, $D_c\left(\lambda \right)$ denotes the dilation of $\lambda$ by $c\ne 0$, ${\mathbb{B}}_r\left(\lambda \right)={\left({\lambda }^{⊞1+r}\right)}^{\uplus {\displaystyle \frac{1}{1+r}}}$ and ${\mathbb{M}}_t\left(\lambda \right)={\left({\lambda }^{⊠(t+1)}\right)}^{\cup \times \frac{1}{t+1}}$. Several examples are presented for important measures $\lambda$ in free probability. Section 2 covers fundamental principles of CSK families and the associated pseudo-variance functions. Section 3 presents the article’s main results and some illustrations are given in Section 4.

2. Cauchy–Stieltjes Kernel Families

In free probability, the CSK families (of probability measures) are newly presented using the CSK ${(1-\theta y)}^{-1}$. Some characteristics of CSK families are presented in [25] using probabilities with compact support. Extended work on CSK families is presented in [26], covering probabilities with support bounded from one side, say from above. Denote ${\mathcal{P}}_{ba}$ as the set of non-degenerate probabilities with support bounded from above. Let $\sigma \in {\mathcal{P}}_{ba}$. With $1/{\theta }_{+}\left(\sigma \right)=max\{\sup \mathrm{supp}\left(\sigma \right),0\}$, the function

$${\mathbf{M}}_{\sigma }\left(\theta \right)=\int {\displaystyle \frac{1}{1-\theta y}}\sigma \left(dy\right)$$

is defined ∀$\theta \in [0,{\theta }_{+}\left(\sigma \right))$. The CSK family induced by $\sigma$ is the set of probabilities

$${\mathcal{K}}_{+}\left(\sigma \right)=\left\{{\mathbf{P}}_{(\theta ,\sigma )}\left(dy\right)={\displaystyle \frac{1}{{\mathbf{M}}_{\sigma }\left(\theta \right)(1-\theta y)}}\sigma \left(dy\right):\theta \in (0,{\theta }_{+}\left(\sigma \right))\right\}.$$

The map $\theta \mapsto {\mathbf{K}}_{\sigma }\left(\theta \right)=\int y{\mathbf{P}}_{(\theta ,\sigma )}\left(dy\right)$ realizes a bijection between $(0,{\theta }_{+}\left(\sigma \right))$ and its image $(m_1\left(\sigma \right),m_{+}\left(\sigma \right))$, which is said to be the mean domain of ${\mathcal{K}}_{+}\left(\sigma \right)$. The inverse function of ${\mathbf{K}}_{\sigma }(·)$ is denoted by ${\chi }_{\sigma }(·)$. For $m\in (m_1\left(\sigma \right),m_{+}\left(\sigma \right))$, write ${\mathbf{Q}}_{(m,\sigma )}\left(dy\right)={\mathbf{P}}_{({\chi }_{\sigma }\left(m\right),\sigma )}\left(dy\right)$, then the mean-parametrization of ${\mathcal{K}}_{+}\left(\sigma \right)$ is

$${\mathcal{K}}_{+}\left(\sigma \right)=\{{\mathbf{Q}}_{(m,\sigma )}\left(dy\right):m\in (m_1\left(\sigma \right),m_{+}\left(\sigma \right))\}.$$

From [26], we know that

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

with

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

If $\sigma$ has a one-sided support boundary from below, the CSK family is denoted ${\mathcal{K}}_{-}\left(\sigma \right)$ and $\theta \in ({\theta }_{-}\left(\sigma \right),0)$ with ${\theta }_{-}\left(\sigma \right)$ is either $1/\mathcal{A}\left(\sigma \right)$ or $-\infty$ and $\mathcal{A}=\mathcal{A}\left(\sigma \right)=min\{\inf supp(\sigma ),0\}$. The mean domain for ${\mathcal{K}}_{-}\left(\sigma \right)$ is $(m_{-}\left(\sigma \right),m_1\left(\sigma \right))$ with $m_{-}\left(\sigma \right)=\mathcal{A}-1/{\mathbf{G}}_{\sigma }\left(\mathcal{A}\right)$. If $\sigma$ has compact support, then $\mathcal{K}\left(\sigma \right)={\mathcal{K}}_{-}\left(\sigma \right)\cup \left\{\sigma \right\}\cup {\mathcal{K}}_{+}\left(\sigma \right)$ is the two-sided CSK family.

The variance of ${\mathbf{Q}}_{(m,\sigma )}$ is denoted as ${\mathbf{V}}_{\sigma }\left(m\right)$ and the function defined on $(m_1\left(\sigma \right),m_{+}\left(\sigma \right))$ by

$$m\mapsto {\mathbf{V}}_{\sigma }\left(m\right)=\int {(y-m)}^2{\mathbf{Q}}_{(m,\sigma )}\left(dy\right)$$

is the variance function (VF) of ${\mathcal{K}}_{+}\left(\sigma \right)$. Unfortunately, if the moment of order 1 of $\sigma \in {\mathcal{P}}_{ba}$ does not exists, then all probabilities in ${\mathcal{K}}_{+}\left(\sigma \right)$ have infinite variance. This issue has led the authors in [26] to define a notion of pseudo-variance function (PVF) as

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

If $-\infty <m_1\left(\sigma \right)=\int y\sigma \left(dy\right)<+\infty$, then the VF exists and (see [26])

$${\mathbb{V}}_{\sigma }\left(m\right)={\displaystyle \frac{m}{m-m_1\left(\sigma \right)}}{\mathbf{V}}_{\sigma }\left(m\right).$$

Next, we review some facts useful in the demonstration of the article’s main results.

Remark 1.

(i) 

Let $\sigma \in {\mathcal{P}}_{ba}$. Consider $g:y\mapsto \kappa y+\beta$, where $\kappa \ne 0$ and $\beta \in \mathbb{R}$ and let $g\left(\sigma \right)$ be the image of σ by g. Then, ∀m is sufficiently close to $m_1\left(g\left(\sigma \right)\right)=g\left(m_1\left(\sigma \right)\right)=\kappa m_1\left(\sigma \right)+\beta$,

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

(2)

(ii) 

${\mathbb{V}}_{\sigma }(·)$ characterizes σ: If we consider ${\rm Y}={\rm Y}\left(m\right)=m+{\displaystyle \frac{{\mathbb{V}}_{\sigma }\left(m\right)}{m}},$ then

$${\mathbf{G}}_{\sigma }\left({\rm Y}\right)={\displaystyle \frac{m}{{\mathbb{V}}_{\sigma }\left(m\right)}}.$$

(3)

(iii) 

Let $\sigma \in {\mathcal{P}}_{ba}$. For $s>0$, so that ${\sigma }^{⊞s}$ is defined and ∀m close sufficiently to $m_1\left({\sigma }^{⊞s}\right)=s{m}_1\left(\sigma \right)$ (see [26])

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

(4)

In addition, ∀$r>0$ and ∀m are sufficiently close to $m_1\left({\sigma }^{\uplus r}\right)=r{m}_1\left(\sigma \right)$ (see [27])

$${\mathbb{V}}_{{\sigma }^{\uplus r}}\left(m\right)=r{\mathbb{V}}_{\sigma }(m/r)+m^2(1/r-1),$$

(5)

Furthermore, ∀m is sufficiently close to $m_1\left({\mathbb{B}}_r\left(\sigma \right)\right)=m_1\left(\sigma \right)$,

$${\mathbb{V}}_{{\mathbb{B}}_r\left(\sigma \right)}\left(m\right)={\mathbb{V}}_{\sigma }\left(m\right)+r{m}^2.$$

(6)

(iv) 

Let $\lambda \in {\mathcal{P}}_{+}$. For $q>0$, so that ${\lambda }^{⊠q}$ is defined and ∀$m\in (m_{-}\left({\lambda }^{⊠q}\right),m_1\left({\lambda }^{⊠q}\right))=({\left(m_{-}\left(\lambda \right)\right)}^q,{\left(m_1\left(\lambda \right)\right)}^q)$, (see [28])

$${\mathbb{V}}_{{\lambda }^{⊠q}}\left(m\right)=m^{2-2/q}{\mathbb{V}}_{\lambda }\left(m^{1/q}\right).$$

(7)

Also, ∀$\alpha \in [0,1]$ and ∀$m\in (m_{-}\left({\lambda }^{\cup \times \alpha }\right),m_1\left({\lambda }^{\cup \times \alpha }\right))=({\left(m_{-}\left(\lambda \right)\right)}^{\alpha },{\left(m_1\left(\lambda \right)\right)}^{\alpha })$, (see [29])

$${\mathbb{V}}_{{\lambda }^{\cup \times \alpha }}\left(m\right)=m^{1-1/\alpha }{\mathbb{V}}_{\lambda }\left(m^{1/\alpha }\right)+m^{1+1/\alpha }-m^2.$$

(8)

In addition, ∀$t>0$ and ∀$m\in (m_{-}\left({\mathbb{M}}_t\left(\lambda \right)\right),m_1\left({\mathbb{M}}_t\left(\lambda \right)\right))=(m_{-}\left(\lambda \right),m_1\left(\lambda \right))$,

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

(9)

Remark 2.

Let $\nu \in \mathcal{P}$. For $s>0$, so that ${\nu }^{⊞s}$ is defined, an injective analytic function exists (said subordination function) ${\mathbf{W}}_s:{\mathbb{C}}^{+}\to {\mathbb{C}}^{+}$, so that ${\mathbf{G}}_{{\nu }^{⊞s}}\left(\xi \right)={\mathbf{G}}_{\nu }\left({\mathbf{W}}_s\left(\xi \right)\right)$, for $\xi \in {\mathbb{C}}^{+}$. In addition,

$${\mathbf{W}}_s\left(\xi \right)=\xi /s+{\displaystyle \frac{(1-1/s)}{{\mathbf{G}}_{{\nu }^{⊞s}}\left(\xi \right)}}$$

(10)

and ${\mathbf{H}}_s\left({\mathbf{W}}_s\left(\xi \right)\right)=\xi$, where

$${\mathbf{H}}_s\left(\xi \right)=s\xi +{\displaystyle \frac{(1-s)}{{\mathbf{G}}_{\nu }\left(\xi \right)}}.$$

(11)

See ([30], Theorem 2.5).

3. Main Results

We present some new limiting measures connected to the FMLLN and involving Boolean and free (additive and multiplicative) convolutions. More precisely, we have:

Theorem 1.

Let λ be an absolutely continuous measure in ${\mathcal{P}}_{+}$. With the notations introduced above, ∀$\alpha \in [0,1]$, ∀$s>1$, ∀$t>0$ and ∀$r>0$,

(i) 

$$\mathsf{\Phi }\left[{\lambda }^{\cup \times \alpha }\right]\left(dm\right)={\left({\displaystyle \frac{1}{m^{1/\alpha -1}-1+m^{-1/\alpha -1}{\mathbb{V}}_{\lambda }\left(m^{1/\alpha }\right)}}\right)}^{\prime }{\mathsf{1}}_{\left({\left(m_{-}\left(\lambda \right)\right)}^{\alpha },{\left(m_1\left(\lambda \right)\right)}^{\alpha }\right)}\left(m\right)dm.$$

(12)

(ii) 

$$\mathsf{\Phi }\left[{\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)}^{\cup \times \alpha }\right]\left(dm\right)={\left({\displaystyle \frac{1}{m^{1/\alpha -1}-1+\frac{m^{-1/\alpha -1}}{s}{\mathbb{V}}_{\lambda }\left(m^{1/\alpha }\right)}}\right)}^{\prime }{\mathsf{1}}_{\left({\left(-{\displaystyle \frac{{\mathbf{W}}_s\left(0\right)}{s-1}}\right)}^{\alpha },{\left(m_1\left(\lambda \right)\right)}^{\alpha }\right)}\left(m\right)dm.$$

(13)

(iii) 

$$\mathsf{\Phi }\left[{\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)}^{\cup \times \alpha }\right]\left(dm\right)={\left({\displaystyle \frac{r}{m^{1/\alpha -1}-r+m^{-1-1/\alpha }{\mathbb{V}}_{\lambda }\left(m^{1/\alpha }\right)}}\right)}^{\prime }{\mathsf{1}}_{\left({\left(m_{-}\left(\lambda \right)\right)}^{\alpha },{\left(m_1\left(\lambda \right)\right)}^{\alpha }\right)}\left(m\right)dm.$$

(14)

(iv) 

$$\mathsf{\Phi }\left[{\left({\mathbb{B}}_r\left(\lambda \right)\right)}^{\cup \times \alpha }\right]\left(dm\right)={\left({\displaystyle \frac{1}{(r+1)m^{1/\alpha -1}-1+m^{-1/\alpha -1}{\mathbb{V}}_{\lambda }\left(m^{1/\alpha }\right)}}\right)}^{\prime }{\mathsf{1}}_{\left({\left(-{\displaystyle \frac{{\mathbf{W}}_{1+r}\left(0\right)}{r}}\right)}^{\alpha },{\left(m_1\left(\lambda \right)\right)}^{\alpha }\right)}\left(m\right)dm.$$

(15)

(v) 

$$\mathsf{\Phi }\left[{\mathbb{M}}_t\left(\lambda \right)\right]\left(dm\right)={\left({\displaystyle \frac{1}{m^{t-2}{\mathbb{V}}_{\lambda }\left(m\right)+m^t-1}}\right)}^{\prime }{\mathsf{1}}_{(m_{-}\left(\lambda \right),m_1\left(\lambda \right))}\left(m\right)dm.$$

(16)

(vi) 

$$\mathsf{\Phi }\left[{\mathbb{M}}_t\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)\right]\left(dm\right)={\left({\displaystyle \frac{s}{m^{t-2}{\mathbb{V}}_{\lambda }\left(m\right)+s(m^t-1)}}\right)}^{\prime }{\mathsf{1}}_{\left(-{\displaystyle \frac{{\mathbf{W}}_s\left(0\right)}{s-1}},m_1\left(\lambda \right)\right)}\left(m\right)dm.$$

(17)

(vii) 

$$\mathsf{\Phi }\left[{\mathbb{M}}_t\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)\right]\left(dm\right)={\left({\displaystyle \frac{r}{m^{t-2}{\mathbb{V}}_{\lambda }\left(m\right)+m^t-r}}\right)}^{\prime }{\mathsf{1}}_{\left(m_{-}\left(\lambda \right),m_1\left(\lambda \right)\right)}\left(m\right)dm.$$

(18)

(viii) 

$$\mathsf{\Phi }\left[{\mathbb{M}}_t\left({\mathbb{B}}_r\left(\lambda \right)\right)\right]\left(dm\right)={\left({\displaystyle \frac{1}{m^{t-2}{\mathbb{V}}_{\lambda }\left(m\right)+(r+1)m^t-1}}\right)}^{\prime }{\mathsf{1}}_{\left(-{\displaystyle \frac{{\mathbf{W}}_{1+r}\left(0\right)}{r}},m_1\left(\lambda \right)\right)}\left(m\right)dm.$$

(19)

Proof. 

(i) According to ([24], Theorem 3.1), for an absolutely continuous measure λ in ${\mathcal{P}}_{+}$, the FMLLN $\mathsf{\Phi }\left[\lambda \right]$ is provided by

$$\mathsf{\Phi }\left[\lambda \right]\left(dm\right)={\left({\displaystyle \frac{m^2}{{\mathbb{V}}_{\lambda }\left(m\right)}}\right)}^{\prime }{\mathbf{1}}_{(m_{-}\left(\lambda \right),m_1\left(\lambda \right))}\left(m\right)dm.$$

(20)

As $\lambda$ is an absolutely continuous measure on $[0,\infty )$, it is the same for ${\lambda }^{\cup \times \alpha }$. Combining (8) and (20), we obtain ∀$\alpha \in [0,1]$

$$\begin{array}{c}\mathsf{\Phi }\left[{\lambda }^{\cup \times \alpha }\right]\left(dm\right)={\left({\displaystyle \frac{m^2}{{\mathbb{V}}_{{\lambda }^{\cup \times \alpha }}\left(m\right)}}\right)}^{\prime }{\mathbf{1}}_{\left(m_{-}\left({\lambda }^{\cup \times \alpha }\right),m_1\left({\lambda }^{\cup \times \alpha }\right)\right)}\left(m\right)dm\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\left({\displaystyle \frac{1}{m^{1/\alpha -1}-1+m^{-1/\alpha -1}{\mathbb{V}}_{\lambda }\left(m^{1/\alpha }\right)}}\right)}^{\prime }{\mathbf{1}}_{\left({\left(m_{-}\left(\lambda \right)\right)}^{\alpha },{\left(m_1\left(\lambda \right)\right)}^{\alpha }\right)}\left(m\right)dm.\end{array}$$

(ii) As λ is an absolutely continuous measure on $[0,\infty )$, it is the same for ${\lambda }^{⊞s}$. We have $\mathcal{A}=\mathcal{A}\left({\lambda }^{⊞s}\right)=min\{\inf supp\left({\lambda }^{⊞s}\right),0\}=0$. So,

$$m_{-}\left({\lambda }^{⊞s}\right)=\mathcal{A}-1/{\mathbf{G}}_{{\lambda }^{⊞s}}\left(\mathcal{A}\right)=-1/{\mathbf{G}}_{{\lambda }^{⊞s}}\left(0\right).$$

(21)

Relations (21) and (10) imply that ∀$s>1$

$$m_{-}\left({\lambda }^{⊞s}\right)=-{\displaystyle \frac{s{\mathbf{W}}_s\left(0\right)}{s-1}}.$$

(22)

Then,

$$m_{-}\left({\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)}^{\cup \times \alpha }\right)={\left(m_{-}\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)\right)}^{\alpha }={\left(-{\displaystyle \frac{{\mathbf{W}}_s\left(0\right)}{s-1}}\right)}^{\alpha }.$$

(23)

Furthermore,

$$m_1\left({\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)}^{\cup \times \alpha }\right)={\left(m_1\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)\right)}^{\alpha }={\left(m_1\left(\lambda \right)\right)}^{\alpha }.$$

(24)

Using (8), (4) and (2), ∀$s>1$ and ∀$\alpha \in [0,1]$ we obtain

$${\mathbb{V}}_{{\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)}^{\cup \times \alpha }}\left(m\right)={\displaystyle \frac{m^{1-1/\alpha }}{s}}{\mathbb{V}}_{\lambda }\left(m^{1/\alpha }\right)+m^{1+1/\alpha }-m^2.$$

(25)

Combining (20), (23)–(25), we obtain ∀$s>1$ and ∀$\alpha \in [0,1]$

$$\mathsf{\Phi }\left[{\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)}^{\cup \times }\alpha \right]\left(dm\right)={\left({\displaystyle \frac{1}{m^{1/\alpha -1}-1+\frac{m^{-1/\alpha -1}}{s}{\mathbb{V}}_{\lambda }\left(m^{1/\alpha }\right)}}\right)}^{\prime }{\mathbf{1}}_{\left({\left(-{\displaystyle \frac{{\mathbf{W}}_s\left(0\right)}{s-1}}\right)}^{\alpha },{\left(m_1\left(\lambda \right)\right)}^{\alpha }\right)}\left(m\right)dm.$$

(iii) As λ is an absolutely continuous measure on $[0,\infty )$, it is the same for ${\lambda }^{\uplus r}$. We have $\mathcal{A}\left({\lambda }^{\uplus r}\right)=min\{\inf supp\left({\lambda }^{\uplus r}\right),0\}=0$. So,

$$m_{-}\left({\lambda }^{\uplus r}\right)=-1/{\mathbf{G}}_{{\lambda }^{\uplus r}}\left(0\right)=-r/{\mathbf{G}}_{\lambda }\left(0\right)=r{m}_{-}\left(\lambda \right).$$

(26)

Then, ∀$r>0$ and ∀$\alpha \in [0,1]$

$$m_{-}\left({\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)}^{\cup \times \alpha }\right)={\left(m_{-}\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)\right)}^{\alpha }={\left(m_{-}\left(\lambda \right)\right)}^{\alpha }.$$

(27)

Furthermore,

$$m_1\left({\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)}^{\cup \times \alpha }\right)={\left(m_1\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)\right)}^{\alpha }={\left(m_1\left(\lambda \right)\right)}^{\alpha }.$$

(28)

Using (8), (5) and (2), ∀$r>0$ and ∀$\alpha \in [0,1]$ we get

$${\mathbb{V}}_{{\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)}^{\cup \times \alpha }}\left(m\right)={\displaystyle \frac{m^{1-1/\alpha }}{r}}{\mathbb{V}}_{\lambda }\left(m^{1/\alpha }\right)+{\displaystyle \frac{m^{1+1/\alpha }}{r}}-m^2.$$

(29)

Combining (20), (27)–(29), we obtain ∀$r>0$ and ∀$\alpha \in [0,1]$

$$\mathsf{\Phi }\left[{\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)}^{\cup \times \alpha }\right]\left(dm\right)={\left({\displaystyle \frac{r}{m^{1/\alpha -1}-r+m^{-1-1/\alpha }{\mathbb{V}}_{\lambda }\left(m^{1/\alpha }\right)}}\right)}^{\prime }{\mathbf{1}}_{\left({\left(m_{-}\left(\lambda \right)\right)}^{\alpha },{\left(m_1\left(\lambda \right)\right)}^{\alpha }\right)}\left(m\right)dm.$$

(iv) Relations (22) and (26) produce

$$m_{-}\left({\mathbb{B}}_r\left(\lambda \right)\right)=m_{-}{\left({\lambda }^{⊞1+r}\right)}^{\uplus {\displaystyle \frac{1}{1+r}}}={\displaystyle \frac{1}{1+r}}{m}_{-}\left({\lambda }^{⊞1+r}\right)=-{\displaystyle \frac{{\mathbf{W}}_{1+r}\left(0\right)}{r}}.$$

Thus, ∀$r>0$ and ∀$\alpha \in [0,1]$ we have

$$m_{-}\left({\left({\mathbb{B}}_r\left(\lambda \right)\right)}^{\cup \times \alpha }\right)={\left(m_{-}\left({\mathbb{B}}_r\left(\lambda \right)\right)\right)}^{\alpha }={\left(-{\displaystyle \frac{{\mathbf{W}}_{1+r}\left(0\right)}{r}}\right)}^{\alpha }.$$

(30)

We also have,

$$m_1\left({\left({\mathbb{B}}_r\left(\lambda \right)\right)}^{\cup \times \alpha }\right)={\left(m_1\left({\mathbb{B}}_r\left(\lambda \right)\right)\right)}^{\alpha }={\left(m_1\left(\lambda \right)\right)}^{\alpha }.$$

(31)

Using (8) and (6), ∀$r>0$, ∀$\alpha \in [0,1]$, and ∀$m<{\left(m_1\left(\lambda \right)\right)}^{\alpha }$ close enough to ${\left(m_1\left(\lambda \right)\right)}^{\alpha }$, we have

$${\mathbb{V}}_{\left({\left({\mathbb{B}}_r\left(\lambda \right)\right)}^{\cup \times \alpha }\right)}\left(m\right)=(r+1)m^{1/\alpha +1}-m^2+m^{-1/\alpha +1}{\mathbb{V}}_{\lambda }\left(m^{1/\alpha }\right)$$

(32)

Combining (20), (30)–(32), we obtain ∀$r>0$ and ∀$\alpha \in [0,1]$

$$\mathsf{\Phi }\left[{\left({\mathbb{B}}_r\left(\lambda \right)\right)}^{\cup \times }\alpha \right]\left(dm\right)={\left({\displaystyle \frac{1}{(r+1)m^{1/\alpha -1}-1+m^{-1/\alpha -1}{\mathbb{V}}_{\lambda }\left(m^{1/\alpha }\right)}}\right)}^{\prime }{\mathbf{1}}_{\left({\left(-{\displaystyle \frac{{\mathbf{W}}_{1+r}\left(0\right)}{r}}\right)}^{\alpha },{\left(m_1\left(\lambda \right)\right)}^{\alpha }\right)}\left(m\right)dm.$$

(v) Relation (16) follows by combining (20) and (9).

(vi) ∀$t>0$ and ∀$s>1$, we have

$$m_{-}\left({\mathbb{M}}_t\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)\right)=m_{-}\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)=-{\displaystyle \frac{{\mathbf{W}}_s\left(0\right)}{s-1}},$$

(33)

and

$$m_1\left({\mathbb{M}}_t\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)\right)=m_1\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)=m_1\left(\lambda \right).$$

(34)

Furthermore, by combining (9), (4) and (2), we obtain

$${\mathbb{V}}_{{\mathbb{M}}_t\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)}\left(m\right)={\displaystyle \frac{m^t}{s}}{\mathbb{V}}_{\lambda }\left(m\right)+m^2(m^t-1).$$

(35)

Relation (17) follows by combining (20), (33)–(35).

(vii) ∀$t>0$ and ∀$r>0$, we have

$$m_{-}\left({\mathbb{M}}_t\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)\right)=m_{-}\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)=m_{-}\left(\lambda \right),$$

(36)

and

$$m_1\left({\mathbb{M}}_t\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)\right)=m_1\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)=m_1\left(\lambda \right).$$

(37)

Furthermore, by combining (9), (5) and (2), we obtain

$${\mathbb{V}}_{{\mathbb{M}}_t\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)}\left(m\right)={\displaystyle \frac{m^t}{r}}{\mathbb{V}}_{\lambda }\left(m\right)+m^2(m^t/r-1).$$

(38)

Relation (18) follows by combining (20), (36)–(38).

(viii) ∀$t>0$ and ∀$r>0$, we have

$$m_{-}\left({\mathbb{M}}_t\left({\mathbb{B}}_r\left(\lambda \right)\right)\right)=m_{-}\left({\mathbb{B}}_r\left(\lambda \right)\right)=-{\displaystyle \frac{{\mathbf{W}}_{1+r}\left(0\right)}{r}},$$

(39)

and

$$m_1\left({\mathbb{M}}_t\left({\mathbb{B}}_r\left(\lambda \right)\right)\right)=m_1\left({\mathbb{B}}_r\left(\lambda \right)\right)=m_1\left(\lambda \right).$$

(40)

Furthermore, by combining (9) and (6), we obtain

$${\mathbb{V}}_{{\mathbb{M}}_t\left({\mathbb{B}}_r\left(\lambda \right)\right)}\left(m\right)=m^t{\mathbb{V}}_{\lambda }\left(m\right)+m^2((r+1)m^t-1).$$

(41)

Relation (19) follows by combining (20), (39)–(41). □

4. Examples

We provide some examples of the limiting laws presented in Theorem 1, using important probability measures λ in free probability.

Example 1.

Consider the Semicircle law

$${\displaystyle \gamma \left(dy\right)=\frac{\sqrt{4-y^2}}{2\pi }{\mathsf{1}}_{(-2,2)}\left(y\right)dy,}$$

with $m_1\left(\gamma \right)=0$. We have

$${\mathbf{V}}_{\gamma }\left(m\right)=1={\mathbb{V}}_{\gamma }\left(m\right),\forall m\in (m_{-}\left(\gamma \right),m_1\left(\gamma \right))=(-1,0)and{\mathbf{G}}_{\gamma }\left(w\right)={\displaystyle \frac{1}{2}}\left(w-\sqrt{w^2-4}\right).$$

Consider $g:y\mapsto y+2$. Then,

$${\displaystyle \lambda \left(dy\right)=g\left(\gamma \right)\left(dy\right)=\frac{\sqrt{y(4-y)}}{2\pi }{\mathsf{1}}_{(0,4)}\left(y\right)dy,}$$

with $m_1\left(\lambda \right)=2$. We have

$${\mathbb{V}}_{\lambda }\left(m\right)={\displaystyle \frac{m}{m-2}},\forall m\in (m_{-}\left(\lambda \right),m_1\left(\lambda \right))=(1,2)and{\mathbf{G}}_{\lambda }\left(w\right)={\displaystyle \frac{1}{2}}\left(w-2-\sqrt{w(w-4)}\right).$$

From (11), we have that

$${\mathbf{H}}_s\left(w\right)=sw-{\displaystyle \frac{2(s-1)}{w-2-\sqrt{w(w-4)}}}$$

and, consequently,

$${\mathbf{W}}_s\left(u\right)={\displaystyle \frac{2s-2s^2+u+su-(s-1)\sqrt{4s(s-1)-4su+u^2}}{2s}}$$

∀$u\in \mathbb{C}\setminus (2s-2\sqrt{s},2s+2\sqrt{s})$. Then,

$${\mathbf{W}}_s\left(0\right)=(s-1)\left(-1-\sqrt{{\displaystyle \frac{s-1}{s}}}\right).$$

Then, we have the following:

(i) ∀$\alpha \in [0,1]$,

$$\mathsf{\Phi }\left[{\lambda }^{\cup \times \alpha }\right]\left(dm\right)={\left({\displaystyle \frac{m^{1/\alpha }-2}{(m^{1/\alpha }-2)(m^{1/\alpha -1}-1)+1/m}}\right)}^{\prime }{\mathsf{1}}_{\left(1,2^{\alpha }\right)}\left(m\right)dm.$$

(ii) ∀$s>1$ and ∀$\alpha \in [0,1]$,

$$\mathsf{\Phi }\left[{\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)}^{\cup \times \alpha }\right]\left(dm\right)={\left({\displaystyle \frac{m^{1/\alpha }-2}{(m^{1/\alpha }-2)(m^{1/\alpha -1}-1)+\frac{1}{sm}}}\right)}^{\prime }{\mathsf{1}}_{\left({\left(1+\sqrt{{\displaystyle \frac{s-1}{s}}}\right)}^{\alpha },2^{\alpha }\right)}\left(m\right)dm.$$

(iii) ∀$r>0$ and ∀$\alpha \in [0,1]$,

$$\mathsf{\Phi }\left[{\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)}^{\cup \times \alpha }\right]\left(dm\right)={\left({\displaystyle \frac{r(m^{1/\alpha }-2)}{(m^{1/\alpha }-2)(m^{1/\alpha -1}-r)+1/m}}\right)}^{\prime }{\mathsf{1}}_{\left(1,2^{\alpha }\right)}\left(m\right)dm.$$

(iv) ∀$r>0$ and ∀$\alpha \in [0,1]$,

$$\mathsf{\Phi }\left[{\left({\mathbb{B}}_r\left(\lambda \right)\right)}^{\cup \times \alpha }\right]\left(dm\right)={\left({\displaystyle \frac{(m^{1/\alpha }-2)}{(m^{1/\alpha }-2)((r+1)m^{1/\alpha -1}-1)+1/m}}\right)}^{\prime }{\mathsf{1}}_{\left({\left(1+\sqrt{{\displaystyle \frac{r}{r+1}}}\right)}^{\alpha },2^{\alpha }\right)}\left(m\right)dm.$$

(v) ∀$t>0$,

$$\mathsf{\Phi }\left[{\mathbb{M}}_t\left(\lambda \right)\right]\left(dm\right)={\left({\displaystyle \frac{m-2}{m^{t-1}+(m^t-1)(m-2)}}\right)}^{\prime }{\mathsf{1}}_{\left(1,2\right)}\left(m\right)dm.$$

(vi) ∀$t>0$ and ∀$s>1$,

$$\mathsf{\Phi }\left[{\mathbb{M}}_t\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)\right]\left(dm\right)={\left({\displaystyle \frac{s(m-2)}{m^{t-1}+s(m^t-1)(m-2)}}\right)}^{\prime }{\mathsf{1}}_{\left(1+\sqrt{{\displaystyle \frac{s-1}{s}}},2\right)}\left(m\right)dm.$$

(vii) ∀$t>0$ and ∀$r>0$,

$$\mathsf{\Phi }\left[{\mathbb{M}}_t\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)\right]\left(dm\right)={\left({\displaystyle \frac{r(m-2)}{m^{t-1}+(m^t-r)(m-2)}}\right)}^{\prime }{\mathsf{1}}_{\left(1,2\right)}\left(m\right)dm.$$

(viii) ∀$t>0$ and ∀$r>0$,

$$\mathsf{\Phi }\left[{\mathbb{M}}_t\left({\mathbb{B}}_r\left(\lambda \right)\right)\right]\left(dm\right)={\left({\displaystyle \frac{(m-2)}{m^{t-1}+((r+1)m^t-1)(m-2)}}\right)}^{\prime }{\mathsf{1}}_{\left(1+\sqrt{{\displaystyle \frac{r}{1+r}}},2\right)}\left(m\right)dm.$$

Example 2.

For $0<a^2<1$, the Marchenko-Pastur measure is provided by

$${\displaystyle \gamma \left(dy\right)=\frac{\sqrt{4-{(y-a)}^2}}{2\pi (1+ay)}{\mathsf{1}}_{(a-2,a+2)}\left(y\right)dy.}$$

We have

$${\mathbb{V}}_{\gamma }\left(m\right)=1+am={\mathcal{V}}_{\gamma }\left(m\right),\forall m\in (m_{-}\left(\gamma \right),m_1\left(\gamma \right))=(-1,0)and{\mathbf{G}}_{\gamma }\left(w\right)={\displaystyle \frac{\left(a+w-\sqrt{{(a-w)}^2-4}\right)}{2(1+aw)}}.$$

Consider $f:y\mapsto ay+1$. We have,

$$\lambda \left(dy\right)=f\left(\gamma \right)\left(dy\right)={\displaystyle \frac{\sqrt{\left({(a+1)}^2-y\right)\left(y-{(a-1)}^2\right)}}{2\pi a^{2}y}}{\mathsf{1}}_{({(a-1)}^2,{(a+1)}^2)}\left(y\right)dy,$$

with $m_1\left(\lambda \right)=1$, and

$${\mathbb{V}}_{\lambda }\left(m\right)={\displaystyle \frac{a^2{m}^2}{m-1}}.$$

The Cauchy transform of λ is

$${\mathbf{G}}_{\lambda }\left(w\right)={\displaystyle \frac{1}{a}}{\mathbf{G}}_{\gamma }\left({\displaystyle \frac{w-1}{a}}\right)={\displaystyle \frac{1}{2wa}}\left(a+{\displaystyle \frac{(w-1)}{a}}-\sqrt{{\left(a-{\displaystyle \frac{(w-1)}{a}}\right)}^2-4}\right).$$

For $m>0$, solving $m+{\mathbb{V}}_{\lambda }\left(m\right)/m=0$, one obtains $m=1-a^2$. Relation (3) implies ${\mathbf{G}}_{\lambda }\left(0\right)=-{\displaystyle \frac{1}{1-a^2}}.$ Thus, $m_{-}\left(\lambda \right)=\mathcal{A}\left(\lambda \right)-1/{\mathbf{G}}_{\lambda }\left(\mathcal{A}\left(\lambda \right)\right)=-1/{\mathbf{G}}_{\lambda }\left(0\right)=1-a^2$. So, $(m_{-}\left(\lambda \right),m_1\left(\lambda \right))=(1-a^2,1).$

We have that

$${\mathbf{H}}_s\left(w\right)=sw-{\displaystyle \frac{2wa(s-1)}{\left(a+\frac{(w-1)}{a}-\sqrt{{\left(a-\frac{(w-1)}{a}\right)}^2-4}\right)}}$$

and

$${\mathbf{W}}_s\left(u\right)={\displaystyle \frac{-a^2+s+a^{2}s-s^2+u+su-(s-1)\sqrt{{(s-a^2)}^2-2u(a^2+s)+u^2}}{2s}},$$

∀$u\in \mathbb{C}\setminus (a^2+s-2\sqrt{s{a}^2},a^2+s+2\sqrt{s{a}^2})$. Then

$${\mathbf{W}}_s\left(0\right)=-{\displaystyle \frac{(s-1)(s-a^2)}{s}}.$$

We have:

(i) ∀$\alpha \in [0,1]$,

$$\mathsf{\Phi }\left[{\lambda }^{\cup \times \alpha }\right]\left(dm\right)={\left({\displaystyle \frac{m^{1/\alpha }-1}{(m^{1/\alpha }-1)(m^{1/\alpha -1}-1)+a^2{m}^{1/\alpha -1}}}\right)}^{\prime }{\mathsf{1}}_{\left({(1-a^2)}^{\alpha },1\right)}\left(m\right)dm.$$

(ii) ∀$s>1$ and ∀$\alpha \in [0,1]$,

$$\mathsf{\Phi }\left[{\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)}^{\cup \times \alpha }\right]\left(dm\right)={\left({\displaystyle \frac{m^{1/\alpha }-1}{(m^{1/\alpha }-1)(m^{1/\alpha -1}-1)+\frac{a^2}{s}{m}^{1/\alpha -1}}}\right)}^{\prime }{\mathsf{1}}_{\left({\left({\displaystyle \frac{s-a^2}{s}}\right)}^{\alpha },1\right)}\left(m\right)dm.$$

(iii) ∀$r>0$ and ∀$\alpha \in [0,1]$,

$$\mathsf{\Phi }\left[{\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)}^{\cup \times \alpha }\right]\left(dm\right)={\left({\displaystyle \frac{r(m^{1/\alpha }-1)}{(m^{1/\alpha }-1)(m^{1/\alpha -1}-r)+a^2{m}^{1/\alpha -1}}}\right)}^{\prime }{\mathsf{1}}_{\left({(1-a^2)}^{\alpha },1\right)}\left(m\right)dm.$$

(iv) ∀$r>0$ and ∀$\alpha \in [0,1]$,

$$\mathsf{\Phi }\left[{\left({\mathbb{B}}_r\left(\lambda \right)\right)}^{\cup \times \alpha }\right]\left(dm\right)={\left({\displaystyle \frac{(m^{1/\alpha }-1)}{(m^{1/\alpha }-1)((r+1)m^{1/\alpha -1}-1)+a^2{m}^{1/\alpha -1}}}\right)}^{\prime }{\mathsf{1}}_{\left({\left({\displaystyle \frac{r+1-a^2}{r+1}}\right)}^{\alpha },1\right)}\left(m\right)dm.$$

(v) ∀$t>0$,

$$\mathsf{\Phi }\left[{\mathbb{M}}_t\left(\lambda \right)\right]\left(dm\right)={\left({\displaystyle \frac{m-1}{a^2{m}^t+(m^t-1)(m-1)}}\right)}^{\prime }{\mathbf{1}}_{\left(1-a^2,1\right)}\left(m\right)dm.$$

(vi) ∀$t>0$ and ∀$s>1$,

$$\mathsf{\Phi }\left[{\mathbb{M}}_t\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)\right]\left(dm\right)={\left({\displaystyle \frac{s(m-1)}{a^2{m}^t+s(m^t-1)(m-1)}}\right)}^{\prime }{\mathsf{1}}_{\left({\displaystyle \frac{s-a^2}{s}},1\right)}\left(m\right)dm.$$

(vii) ∀$t>0$ and ∀$r>0$,

$$\mathsf{\Phi }\left[{\mathbb{M}}_t\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)\right]\left(dm\right)={\left({\displaystyle \frac{r(m-1)}{a^2{m}^t+(m^t-r)(m-1)}}\right)}^{\prime }{\mathsf{1}}_{\left(1-a^2,1\right)}\left(m\right)dm.$$

(viii) ∀$t>0$ and ∀$r>0$,

$$\mathsf{\Phi }\left[{\mathbb{M}}_t\left({\mathbb{B}}_r\left(\lambda \right)\right)\right]\left(dm\right)={\left({\displaystyle \frac{(m-1)}{a^2{m}^t+((r+1)m^t-1)(m-1)}}\right)}^{\prime }{\mathsf{1}}_{\left({\displaystyle \frac{r+1-a^2}{r+1}},1\right)}\left(m\right)dm.$$

Example 3.

The free Gamma measure is provided, for $a\ne 0$, by

$$\gamma \left(dy\right)={\displaystyle \frac{\sqrt{4(1+a^2)-{(y-2a)}^2}}{2\pi (a^2{y}^2+2ay+1)}}{\mathsf{1}}_{(-2\sqrt{1+a^2}+2a,2\sqrt{1+a^2}+2a)}\left(y\right)\left(dy\right),$$

with $m_1\left(\gamma \right)=0$. We have

$${\mathbb{V}}_{\gamma }\left(m\right)={\mathcal{V}}_{\gamma }\left(m\right)={(1+am)}^{2}and{\mathbf{G}}_{\gamma }\left(w\right)={\displaystyle \frac{2a+w+2a^{2}w-\sqrt{{(2a-w)}^2-4(1+a^2)}}{2{(1+aw)}^2}}.$$

Consider $h:y\mapsto ay+1$. Then,

$$\lambda \left(dy\right)=h\left(\gamma \right)\left(dy\right)={\displaystyle \frac{\sqrt{({(a+\sqrt{a^2+1})}^2-y)(y-{(-a+\sqrt{a^2+1})}^2)}}{2\pi a^2{y}^2}}{\mathsf{1}}_{\left(\right(\sqrt{a^2+1}-{\left|a\right|)}^2,(\sqrt{a^2+1}+{\left|a\right|)}^2)}\left(y\right)dy,$$

with $m_1\left(\lambda \right)=1$. We have

$${\mathbb{V}}_{\lambda }\left(m\right)={\displaystyle \frac{a^2{m}^3}{m-1}},\forall m\in (m_{-}\left(\lambda \right),m_1\left(\lambda \right))=\left({\displaystyle \frac{1}{1+a^2}},1\right)$$

and

$${\mathbf{G}}_{\lambda }\left(w\right)={\displaystyle \frac{\frac{w-1}{a}+2aw-\sqrt{{(2a-\frac{w-1}{a})}^2-4(1+a^2)}}{2a{w}^2}}.$$

We also have

$${\mathbf{H}}_s\left(w\right)=sw-{\displaystyle \frac{2(s-1)a{w}^2}{\frac{w-1}{a}+2aw-\sqrt{{(2a-\frac{w-1}{a})}^2-4(1+a^2)}}}$$

and

$${\mathbf{W}}_s\left(u\right)={\displaystyle \frac{s-s^2+u(1+2a^2+s)-(s-1)\sqrt{s^2-(4a^2+2s)u+u^2}}{2(a^2+s)}}$$

∀$u\in \mathbb{C}\setminus (2a^2+s-\sqrt{{(2a^2+s)}^2-1},2a^2+s+\sqrt{{(2a^2+s)}^2-1})$. Then,

$${\mathbf{W}}_s\left(0\right)=-{\displaystyle \frac{s(s-1)}{a^2+s}}.$$

We have:

(i) ∀$\alpha \in [0,1]$,

$$\mathsf{\Phi }\left[{\lambda }^{\cup \times \alpha }\right]\left(dm\right)={\left({\displaystyle \frac{m^{1/\alpha }-1}{(m^{1/\alpha }-1)(m^{1/\alpha -1}-1)+a^2{m}^{2/\alpha -1}}}\right)}^{\prime }{\mathsf{1}}_{\left({\left({\displaystyle \frac{1}{1+a^2}}\right)}^{\alpha },1\right)}\left(m\right)dm.$$

(ii) ∀$s>1$ and ∀$\alpha \in [0,1]$,

$$\mathsf{\Phi }\left[{\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)}^{\cup \times \alpha }\right]\left(dm\right)={\left({\displaystyle \frac{m^{1/\alpha }-1}{(m^{1/\alpha }-1)(m^{1/\alpha -1}-1)+\frac{a^2}{s}{m}^{2/\alpha -1}}}\right)}^{\prime }{\mathsf{1}}_{\left({\left({\displaystyle \frac{s}{s+a^2}}\right)}^{\alpha },1\right)}\left(m\right)dm.$$

(iii) ∀$r>0$ and ∀$\alpha \in [0,1]$,

$$\mathsf{\Phi }\left[{\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)}^{\cup \times \alpha }\right]\left(dm\right)={\left({\displaystyle \frac{r(m^{1/\alpha }-1)}{(m^{1/\alpha }-1)(m^{1/\alpha -1}-r)+a^2{m}^{2/\alpha -1}}}\right)}^{\prime }{\mathsf{1}}_{\left({\left({\displaystyle \frac{1}{1+a^2}}\right)}^{\alpha },1\right)}\left(m\right)dm.$$

(iv) ∀$r>0$ and ∀$\alpha \in [0,1]$,

$$\mathsf{\Phi }\left[{\left({\mathbb{B}}_r\left(\lambda \right)\right)}^{\cup \times \alpha }\right]\left(dm\right)={\left({\displaystyle \frac{(m^{1/\alpha }-1)}{(m^{1/\alpha }-1)((r+1)m^{1/\alpha -1}-1)+a^2{m}^{2/\alpha -1}}}\right)}^{\prime }{\mathsf{1}}_{\left({\left({\displaystyle \frac{r+1}{r+1+a^2}}\right)}^{\alpha },1\right)}\left(m\right)dm.$$

(v) ∀$t>0$,

$$\mathsf{\Phi }\left[{\mathbb{M}}_t\left(\lambda \right)\right]\left(dm\right)={\left({\displaystyle \frac{m-1}{a^2{m}^{t+1}+(m^t-1)(m-1)}}\right)}^{\prime }{\mathsf{1}}_{\left({\displaystyle \frac{1}{1+a^2}},1\right)}\left(m\right)dm.$$

(vi) ∀$t>0$ and ∀$s>1$,

$$\mathsf{\Phi }\left[{\mathbb{M}}_t\left(D_{1/s}\left({\lambda }^{⊞s}\right)\right)\right]\left(dm\right)={\left({\displaystyle \frac{s(m-1)}{a^2{m}^{t+1}+s(m^t-1)(m-1)}}\right)}^{\prime }{\mathsf{1}}_{\left({\displaystyle \frac{s}{s+a^2}},1\right)}\left(m\right)dm.$$

(vii) ∀$t>0$ and ∀$r>0$,

$$\mathsf{\Phi }\left[{\mathbb{M}}_t\left(D_{1/r}\left({\lambda }^{\uplus r}\right)\right)\right]\left(dm\right)={\left({\displaystyle \frac{r(m-1)}{a^2{m}^{t+1}+(m^t-r)(m-1)}}\right)}^{\prime }{\mathsf{1}}_{\left({\displaystyle \frac{1}{1+a^2}},1\right)}\left(m\right)dm.$$

(viii) ∀$t>0$ and ∀$r>0$,

$$\mathsf{\Phi }\left[{\mathbb{M}}_t\left({\mathbb{B}}_r\left(\lambda \right)\right)\right]\left(dm\right)={\left({\displaystyle \frac{(m-1)}{a^2{m}^{t+1}+((r+1)m^t-1)(m-1)}}\right)}^{\prime }{\mathsf{1}}_{\left({\displaystyle \frac{r+1}{r+1+a^2}},1\right)}\left(m\right)dm.$$

5. Conclusions

The law of large numbers (LLN) for the classical additive convolution in classical probability can be used to derive the LLN for the classical multiplicative convolution. In the situation of non-commutative probability, this is not true. For probability measures with bounded support, the free additive law was proven in [31], and it was extended to all measures with finite first moment in [15]. For measures with restricted support, the FMLLN was proven in [13], and it was extended to measures with unbounded support in [14]. Except in the situation where the original measure is a Dirac measure, the limiting measure for the FMLLN is not a Dirac measure, in contrast with the situation of classical multiplicative convolution. Using the variance functions, an intriguing explanation of the FMLLN is provided in [24]. The FMLLN for different kinds of non-commutative probability measures including the Boolean and free (additive and multiplicative) convolutions has been clearly shown in this study. Researchers studying non-commutative probability may find the results helpful as they are explained using a variety of probability metrics. The approach used in this paper provides an explicit expression of the limiting distributions basing on the variance function, which plays a crucial role in the setting of CSK families.