Stability of Cauchy–Stieltjes Kernel Families by Free and Boolean Convolutions Product

Abstract

Let $\mathfrak{F}\left({\nu }_j\right)=\{Q_{m_j}^{{\nu }_j},m_j\in (m_{-}^{{\nu }_j},m_{+}^{{\nu }_j})\}$, $j=1,2$, be two Cauchy–Stieltjes Kernel (CSK) families induced by non-degenerate compactly supported probability measures ${\nu }_1$ and ${\nu }_2$. Introduce the set of measures $\mathcal{F}=\mathfrak{F}\left({\nu }_1\right)⊞\mathfrak{F}\left({\nu }_2\right)=\{Q_{m_1}^{{\nu }_1}⊞Q_{m_2}^{{\nu }_2},m_1\in (m_{-}^{{\nu }_1},m_{+}^{{\nu }_1})and{m}_2\in (m_{-}^{{\nu }_2},m_{+}^{{\nu }_2})\}.$ We show that if $\mathcal{F}$ remains a CSK family, (i.e., $\mathcal{F}=\mathfrak{F}\left(\mu \right)$ where $\mu$ is a non-degenerate compactly supported measure), then the measures $\mu$, ${\nu }_1$ and ${\nu }_2$ are of the Marchenko–Pastur type measure up to affinity. A similar conclusion is obtained if we substitute (in the definition of $\mathcal{F}$) the additive free convolution ⊞ by the additive Boolean convolution ⊎. The cases where the additive free convolution ⊞ is replaced (in the definition of $\mathcal{F}$) by the multiplicative free convolution ⊠ or the multiplicative Boolean convolution ⨃ are also studied.

1. Introduction

Convolution stability is an important concept in probability theory, particularly in the study of probability laws and stochastic processes. It represents a fundamental concept that helps understand how certain probability laws behave under addition and has profound implications in various practical applications.

In [1], the authors studied the stability (with respect to classical convolution) of natural exponential families (NEFs) of probability measures: If the convolution of NEFs on ${\mathbb{R}}^d$ remains a NEF, then the NEFs are all Poisson–Gaussian, up to affinity. This result is a generalization of the one-dimensional versions demonstrated in [2] in the case of two NEFs, and in [3] for more than two NEFs.

In non-commutative probability, the concept of Cauchy–Stieltjes kernel (CSK) families is newly defined. It concerns families of probability measures introduced in a manner similar to that of NEFs by involving the Cauchy–Stieltjes kernel (CSK) ${\displaystyle \frac{1}{1-\vartheta y}}$ instead of the exponential kernel $exp\left(\vartheta y\right)$. The CSK families have been studied in [4] for measures with compact support. Further properties are proven in [5], by exploring measures with a one-sided support boundary, such as the one mentioned above. In this article, we study the stability of CSK families with respect to free and Boolean convolutions product. This stability property subsumes the CSK family induced by the Marchenko–Pastur type law and provides some new variance functions with non-usual expressions. The approach used in this article extends the understanding of CSK families and their stability in the non-commutative probability framework. In the rest of this section, after giving some basic concepts about CSK families, the purpose of this article is presented in more detail.

Let $\mu$ be a (non-degenerate) probability measure with a one-sided support boundary from above. Then,

$${\mathbb{M}}_{\mu }\left(\vartheta \right)=\int {\displaystyle \frac{1}{1-\vartheta y}}\mu \left(dy\right)$$

converges ∀$\vartheta \in [0,{\vartheta }_{+}^{\mu })$ with $1/{\vartheta }_{+}^{\mu }=max\{sup\mathrm{supp}\left(\mu \right),0\}$. The CSK family induced by $\mu$ is the family of probabilities

$${\mathfrak{F}}_{+}\left(\mu \right)=\{{\mathbb{P}}_{(\vartheta ,\mu )}\left(dy\right)={\displaystyle \frac{1}{{\mathbb{M}}_{\mu }\left(\vartheta \right)(1-\vartheta y)}}\mu \left(dy\right):\vartheta \in (0,{\vartheta }_{+}^{\mu })\}.$$

The map $\vartheta \mapsto {\mathbb{K}}_{\mu }\left(\vartheta \right)=\int y{\mathbb{P}}_{(\vartheta ,\mu )}\left(dy\right)$ represent mean function. It is increasing strictly on $(0,{\vartheta }_{+}^{\mu })$, see [5]. The interval $(m_0^{\mu },m_{+}^{\mu })={\mathbb{K}}_{\mu }\left((0,{\vartheta }_{+}^{\mu })\right)$ is said to be the mean domain of ${\mathfrak{F}}_{+}\left(\mu \right)$. This leads to the mean re-parametrization of ${\mathfrak{F}}_{+}\left(\mu \right)$. Let ${\psi }_{\mu }(·)$ the inverse function of ${\mathbb{K}}_{\mu }(·)$ and for $x\in (m_0^{\mu },m_{+}^{\mu })$, write $Q_x^{\mu }\left(dy\right)={\mathbb{P}}_{({\psi }_{\mu }\left(x\right),\mu )}\left(dy\right)$. We obtain

$${\mathfrak{F}}_{+}\left(\mu \right)=\{Q_x^{\mu }\left(dy\right):x\in (m_0^{\mu },m_{+}^{\mu })\}.$$

Consider

$$\mathcal{B}=\mathcal{B}\left(\mu \right)=max\{0,sup\mathrm{supp}\left(\mu \right)\}=1/{\vartheta }_{+}\in [0,\infty ).$$

It is known from [5] that

$$m_0^{\mu }=\underset{\vartheta \to 0^{+}}{lim}{\mathbb{K}}_{\mu }\left(\vartheta \right)\mathrm{and}{m}_{+}^{\mu }=\mathcal{B}-\underset{w\to {\mathcal{B}}^{+}}{lim}{\displaystyle \frac{1}{{\mathcal{G}}_{\mu }\left(w\right)}},$$

where

$${\mathcal{G}}_{\mu }\left(w\right)=\int {\displaystyle \frac{1}{w-y}}\mu \left(dy\right),w\in \mathbb{C}\setminus \mathrm{supp}\left(\mu \right).$$

is the Cauchy transform of $\mu$.

The CSK family induced by a measure $\mu$ with a support bounded from below can also be introduced. It is denoted ${\mathfrak{F}}_{-}\left(\mu \right)$ and parameterized by $\vartheta \in ({\vartheta }_{-}^{\mu },0)$, where ${\vartheta }_{-}^{\mu }$ is either $-\infty$ or $1/\mathcal{A}\left(\mu \right)$ with $\mathcal{A}=\mathcal{A}\left(\mu \right)=min\{infsupp(\mu ),0\}$. The mean domain for ${\mathfrak{F}}_{-}\left(\mu \right)$ is $(m_{-}^{\mu },m_0^{\mu })$ with $m_{-}^{\mu }=\mathcal{A}-1/{\mathcal{G}}_{\mu }\left(\mathcal{A}\right)$. If $\mu$ is compactly supported, then $\mathfrak{F}\left(\mu \right)={\mathfrak{F}}_{+}\left(\mu \right)\cup {\mathfrak{F}}_{-}\left(\mu \right)\cup \left\{\mu \right\}$ is the two-sided CSK family.

The variance function (VF) is defined by (see [4])

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

If the first moment of $\mu$ does not exist, then all the laws in ${\mathfrak{F}}_{+}\left(\mu \right)$ have infinite variance. The notion of pseudo-variance function ${\mathbb{V}}_{\mu }(·)$ is defined in [5] as

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

If $m_0^{\mu }=\int y\mu \left(dy\right)$ exists finitely, then the VF exists (see [5]) and

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

Remark 1.

(i) ${\mathbb{V}}_{\mu }(·)$ determine μ: Consider $\varpi =\varpi \left(x\right)=x+{\displaystyle \frac{{\mathbb{V}}_{\mu }\left(x\right)}{x}},$ then

$${\mathcal{G}}_{\mu }\left(\varpi \right)={\displaystyle \frac{x}{{\mathbb{V}}_{\mu }\left(x\right)}}.$$

Furthermore, if $m_0^{\mu }$ is finite, then

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

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

(ii) 

Let $\chi \left(\mu \right)$ be the image of μ by $\chi :y\mapsto \xi y+\iota$ where $\xi \ne 0$ and $\iota \in \mathbb{R}$. Then, ∀m close to $m_0^{\chi \left(\mu \right)}=\chi \left(m_0^{\mu }\right)=\xi m_0^{\mu }+\iota$,

$${\mathbb{V}}_{\chi \left(\mu \right)}\left(x\right)={\displaystyle \frac{{\xi }^{2}x}{x-\iota }}{\mathbb{V}}_{\mu }\left({\displaystyle \frac{x-\iota }{\xi }}\right).$$

If the VF exists, then

$${\mathcal{V}}_{\chi \left(\mu \right)}\left(x\right)={\xi }^2{\mathcal{V}}_{\mu }\left({\displaystyle \frac{x-\iota }{\xi }}\right).$$

(iii) 

For $a\ne 0$, the Marchenko–Pastur law is

$${\displaystyle {\mathbb{MP}}^{\left(a\right)}\left(ds\right)=\frac{\sqrt{4-{(s-a)}^2}}{2\pi (1+as)}{\mathbf{1}}_{(a-2,a+2)}\left(s\right)ds+(1-1/a^2){\delta }_{-1/a}\left(ds\right),}$$

with $m_0^{{\mathbb{MP}}^{\left(a\right)}}=0$. We know that

$${\mathcal{V}}_{{\mathbb{MP}}^{\left(a\right)}}\left(m\right)=1+am,\forall m\in (m_{-}^{{\mathbb{MP}}^{\left(a\right)}},m_{+}^{{\mathbb{MP}}^{\left(a\right)}})=(-1,1).$$

See ([4], [Theorem 3.2]), for more details.

Now, we recall the definition of additive free convolution (see [6] for more details) and we present the purpose of this article. Denote by $\mathcal{P}$ (respectively, by ${\mathcal{P}}^{+}$, ${\mathcal{P}}_c$, ${\mathcal{P}}_c^{+}$) the set of non-degenerate probability measures on $\mathbb{R}$ (respectively, the subset of measures from $\mathcal{P}$ concentrated on the positive real line, with compact support, concentrated on the positive real line and compactly supported). The additive free convolution of $\mu$ and $\nu$$\in \mathcal{P}$ is the measure denoted as $\mu ⊞\nu$, satisfying

$${\mathcal{R}}_{\mu ⊞\nu }\left(\xi \right)={\mathcal{R}}_{\mu }\left(\xi \right)+{\mathcal{R}}_{\nu }\left(\xi \right),$$

where the free cumulant transformation, ${\mathcal{R}}_{\mu }$, of $\mu$ is

$${\mathcal{R}}_{\mu }\left({\mathcal{G}}_{\mu }\left(\xi \right)\right)=\xi -{\displaystyle \frac{1}{{\mathcal{G}}_{\mu }\left(\xi \right)}},\forall \xi \mathrm{clos} \mathrm{to} 0.$$

The objective in this article related to the ⊞-convolution is as follows: Let $\mathfrak{F}\left({\nu }_j\right)=\{Q_{m_j}^{{\nu }_j},m_j\in (m_{-}^{{\nu }_j},m_{+}^{{\nu }_j})\}$, $j=1,2$, be two CSK families induced by ${\nu }_1$ and ${\nu }_2\in {\mathcal{P}}_c$. Introduce the family of probabilities

$$\mathcal{F}=\mathfrak{F}\left({\nu }_1\right)⊞\mathfrak{F}\left({\nu }_2\right)=\{Q_{m_1}^{{\nu }_1}⊞Q_{m_2}^{{\nu }_2},m_1\in (m_{-}^{{\nu }_1},m_{+}^{{\nu }_1})\mathrm{and}{m}_2\in (m_{-}^{{\nu }_2},m_{+}^{{\nu }_2})\}.$$

We demonstrate that if $\mathcal{F}$ remains a CSK family, (i.e., $\mathcal{F}=\mathfrak{F}\left(\mu \right)$ for $\mu \in {\mathcal{P}}_c$), then the measures $\mu$, ${\nu }_1$ and ${\nu }_2$ are of the Marchenko–Pastur type law up to affinity. We obtain the same conclusion if we replace (in the definition of $\mathcal{F}$) the additive free convolution ⊞ by the additive Boolean convolution ⊎. The cases where the additive free convolution ⊞ is replaced (in the definition of $\mathcal{F}$) by the multiplicative free convolution ⊠ or the multiplicative Boolean convolution ⨃ are also studied. This provides some new VFs with non-usual forms.

2. Main Results

2.1. Stability of CSK Families by Free Additive Convolution Product

Theorem 1.

Let ${\nu }_1$, ${\nu }_2\in {\mathcal{P}}_c$. Introduce the family of measures

$$\mathcal{F}=\mathfrak{F}\left({\nu }_1\right)⊞\mathfrak{F}\left({\nu }_2\right)=\{Q_{m_1}^{{\nu }_1}⊞Q_{m_2}^{{\nu }_2},m_1\in (m_{-}^{{\nu }_1},m_{+}^{{\nu }_1})and{m}_2\in (m_{-}^{{\nu }_2},m_{+}^{{\nu }_2})\}.$$

If $\mathcal{F}$ remains a CSK family, (i.e., $\mathcal{F}=\mathfrak{F}\left(\mu \right)$ for $\mu \in {\mathcal{P}}_c$), then μ, ${\nu }_1$ and ${\nu }_2$ are of the Marchenko–Pastur type law up to affinity.

Proof. 

We omit the proof because it is the same when considering $t=1$ in the proof of ([7], [Theorem 2]). □

2.2. Stability of CSK Families by Boolean Additive Convolution Product

For $\mu$, $\nu \in \mathcal{P}$, the additive Boolean convolution $\mu \uplus \nu$ is the probability measure introduced by

$${\mathcal{E}}_{\mu \uplus \nu }\left(\xi \right)={\mathcal{E}}_{\mu }\left(\xi \right)+{\mathcal{E}}_{\nu }\left(\xi \right),\mathrm{for}\xi \in {\mathbb{C}}^{+},$$

where

$${\mathcal{E}}_{\mu }\left(\xi \right)=\xi -{\displaystyle \frac{1}{{\mathcal{G}}_{\mu }\left(\xi \right)}},\mathrm{for}\xi \in {\mathbb{C}}^{+}.$$

See [8] for more details.

Theorem 2.

Let ${\nu }_1$, ${\nu }_2\in {\mathcal{P}}_c$. Introduce the family of probabilities

$$\mathcal{H}=\mathfrak{F}\left({\nu }_1\right)\uplus \mathfrak{F}\left({\nu }_2\right)=\{Q_{m_1}^{{\nu }_1}\uplus Q_{m_2}^{{\nu }_2},m_1\in (m_{-}^{{\nu }_1},m_{+}^{{\nu }_1})and{m}_2\in (m_{-}^{{\nu }_2},m_{+}^{{\nu }_2})\}.$$

If $\mathcal{H}$ remains a CSK family, (i.e., $\mathcal{H}=\mathfrak{F}\left(\sigma \right)$ for $\sigma \in {\mathcal{P}}_c$), then σ, ${\nu }_1$ and ${\nu }_2$ are of the Marchenko–Pastur type law up to affinity.

Proof. 

Assume that $\mathcal{H}=\mathfrak{F}\left(\sigma \right)$ for $\sigma \in {\mathcal{P}}_c$, (we may suppose, without loss of generality, that $0\in (m_{-}^{\sigma },m_{+}^{\sigma })$). Then, ∀$m_1\in (m_{-}^{{\nu }_1},m_{+}^{{\nu }_1})$ and ∀$m_2\in (m_{-}^{{\nu }_2},m_{+}^{{\nu }_2})$, there is $t\in (m_{-}^{\sigma },m_{+}^{\sigma })$, so that

$$Q_t^{\sigma }=Q_{m_1}^{{\nu }_1}\uplus Q_{m_2}^{{\nu }_2}.$$

Or equivalently, ∀$\xi$ large enough

$${\mathcal{E}}_{Q_t^{\sigma }}\left(\xi \right)={\mathcal{E}}_{Q_{m_1}^{{\nu }_1}}\left(\xi \right)+{\mathcal{E}}_{Q_{m_2}^{{\nu }_2}}\left(\xi \right).$$

(1)

For $\sigma \in {\mathcal{P}}_c$, the Boolean cumulant transform of $Q_t^{\sigma }\in {\mathcal{P}}_c$ may be expressed as

$${\mathcal{E}}_{Q_t^{\sigma }}\left(\xi \right)=r_1\left(Q_t^{\sigma }\right)+r_2\left(Q_t^{\sigma }\right){\displaystyle \frac{1}{\xi }}+{\displaystyle \frac{1}{\xi }}\epsilon \left({\displaystyle \frac{1}{\xi }}\right)\mathrm{with}\epsilon \left({\displaystyle \frac{1}{\xi }}\right)\stackrel{\xi \to +\infty }{\to }0.$$

(2)

where $r_1\left(Q_t^{\sigma }\right)$ and $r_2\left(Q_t^{\sigma }\right)$ denotes, respectively, the first and the second Boolean cumulants of $Q_t^{\sigma }$. It follows from ([9], [Equation 3.2]) that

$$r_1\left(Q_t^{\sigma }\right)=t\mathrm{and}{r}_2\left(Q_t^{\sigma }\right)=\mathrm{variance} \mathrm{of}{Q}_t^{\sigma }={\mathcal{V}}_{\sigma }\left(t\right).$$

So, relation (2) reduces to

$${\mathcal{E}}_{Q_t^{\sigma }}\left(\xi \right)=t+{\mathcal{V}}_{\sigma }\left(t\right){\displaystyle \frac{1}{\xi }}+{\displaystyle \frac{1}{\xi }}\epsilon \left({\displaystyle \frac{1}{\xi }}\right)\mathrm{with}\epsilon \left({\displaystyle \frac{1}{\xi }}\right)\stackrel{\xi \to +\infty }{\to }0.$$

(3)

Similarly, the Boolean cumulant transforms of $Q_{m_1}^{{\nu }_1}$ and $Q_{m_2}^{{\nu }_2}\in {\mathcal{P}}_c$ may be written as

$${\mathcal{E}}_{Q_{m_1}^{{\nu }_1}}\left(\xi \right)=m_1+{\mathcal{V}}_{{\nu }_1}\left(m_1\right){\displaystyle \frac{1}{\xi }}+{\displaystyle \frac{1}{\xi }}{\epsilon }_1\left({\displaystyle \frac{1}{\xi }}\right)\mathrm{with}{\epsilon }_1\left({\displaystyle \frac{1}{\xi }}\right)\stackrel{\xi \to +\infty }{\to }0$$

(4)

and

$${\mathcal{E}}_{Q_{m_2}^{{\nu }_2}}\left(\xi \right)=m_2+{\mathcal{V}}_{{\nu }_2}\left(m_2\right){\displaystyle \frac{1}{\xi }}+{\displaystyle \frac{1}{\xi }}{\epsilon }_2\left({\displaystyle \frac{1}{\xi }}\right)\mathrm{with}{\epsilon }_2\left({\displaystyle \frac{1}{\xi }}\right)\stackrel{\xi \to +\infty }{\to }0.$$

(5)

Equation (1), together with (3), (4), (5) implies that

$$t+{\mathcal{V}}_{\sigma }\left(t\right){\displaystyle \frac{1}{\xi }}+{\displaystyle \frac{1}{\xi }}\epsilon \left({\displaystyle \frac{1}{\xi }}\right)=m_1+m_2+({\mathcal{V}}_{{\nu }_1}\left(m_1\right)+{\mathcal{V}}_{{\nu }_2}\left(m_2\right)){\displaystyle \frac{1}{\xi }}+{\displaystyle \frac{1}{\xi }}\left({\epsilon }_1\left({\displaystyle \frac{1}{\xi }}\right)+{\epsilon }_2\left({\displaystyle \frac{1}{\xi }}\right)\right).$$

(6)

When $\xi \to +\infty$ in (6), we obtain $t=m_1+m_2$. So, Equation (6) is reduced to

$${\mathcal{V}}_{\sigma }(m_1+m_2){\displaystyle \frac{1}{\xi }}+{\displaystyle \frac{1}{\xi }}\epsilon \left({\displaystyle \frac{1}{\xi }}\right)=({\mathcal{V}}_{{\nu }_1}\left(m_1\right)+{\mathcal{V}}_{{\nu }_2}\left(m_2\right)){\displaystyle \frac{1}{\xi }}+{\displaystyle \frac{1}{\xi }}\left({\epsilon }_1\left({\displaystyle \frac{1}{\xi }}\right)+{\epsilon }_2\left({\displaystyle \frac{1}{\xi }}\right)\right).$$

(7)

Multiplying by $\xi$ in both sides of (7) and letting $\xi \to +\infty$, we obtain

$${\mathcal{V}}_{\sigma }(m_1+m_2)={\mathcal{V}}_{{\nu }_1}\left(m_1\right)+{\mathcal{V}}_{{\nu }_2}\left(m_2\right).$$

According to [10], we obtain

$${\mathcal{V}}_{\sigma }\left(m\right)={\mathcal{V}}_{{\nu }_1}\left(m\right)=\gamma m+\varsigma \mathrm{and}{\mathcal{V}}_{{\nu }_2}\left(m\right)=\gamma m\mathrm{for}\gamma \ne 0\mathrm{and}\varsigma >0.$$

(8)

The parameter $\varsigma ={\mathcal{V}}_{\sigma }\left(0\right)$ should be strictly positive. In addition, $m_0^{{\nu }_2}\ne 0$. Otherwise $m\mapsto {\mathcal{V}}_{{\nu }_2}\left(m\right)=\lambda m$ cannot be a VF, see ([11], [page 6]). Clearly from (8), measures $\sigma$, ${\nu }_1$ and ${\nu }_2$ are of Marchenko–Pastur type law up to affinity. □

2.3. Stability of CSK Families by Multiplicative Free Convolution Product

For $\mu \in {\mathcal{P}}^{+}$, the $\mathcal{S}$-transform is defined by

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

The product of $\mathcal{S}$-transforms is an $\mathcal{S}$-transform and defines, for ${\nu }_1$, ${\nu }_2\in {\mathcal{P}}^{+}$, the multiplicative free convolution ${\nu }_1⊠{\nu }_2$, by ${\mathcal{S}}_{{\nu }_1⊠{\nu }_2}\left(z\right)={\mathcal{S}}_{{\nu }_1}\left(z\right){\mathcal{S}}_{{\nu }_2}\left(z\right)$. See [6,12] for more details.

Theorem 3.

Let ${\nu }_1$, ${\nu }_2\in {\mathcal{P}}_c^{+}$. Introduce the family of probabilities

$$\mathcal{Z}=\mathfrak{F}\left({\nu }_1\right)⊠\mathfrak{F}\left({\nu }_2\right)=\{Q_{m_1}^{{\nu }_1}⊠Q_{m_2}^{{\nu }_2},m_1\in (m_{-}^{{\nu }_1},m_{+}^{{\nu }_1})and{m}_2\in (m_{-}^{{\nu }_2},m_{+}^{{\nu }_2})\}.$$

If $\mathcal{Z}$ remains a CSK family, (i.e., $\mathcal{Z}=\mathfrak{F}\left(\sigma \right)$ for $\sigma \in {\mathcal{P}}_c^{+}$), then σ, ${\nu }_1$ and ${\nu }_2$ are such that

$${\mathcal{V}}_{{\nu }_1}\left(m\right)=m^2(cln\left(m\right)+a_1),\forall m\in (m_{-}^{{\nu }_1},m_{+}^{{\nu }_2}),$$

(9)

$${\mathcal{V}}_{{\nu }_2}\left(m\right)=m^2(cln\left(m\right)+a_2),\forall m\in (m_{-}^{{\nu }_2},m_{+}^{{\nu }_2}),$$

(10)

and

$${\mathcal{V}}_{\sigma }\left(m\right)=m^2(cln\left(m\right)+a_1+a_2),\forall m\in (m_{-}^{\sigma },m_{+}^{\sigma }).$$

(11)

$a_1$ and $a_2$ are positive strictly and $c\in (\kappa ,+\infty )$ with $\kappa =max\{{\displaystyle \frac{-a_1}{ln\left(m_{+}^{{\nu }_1}\right)}},{\displaystyle \frac{-a_2}{ln\left(m_{+}^{{\nu }_2}\right)}},{\displaystyle \frac{-(a_1+a_2)}{ln\left(m_{+}^{\sigma }\right)}}\}$.

Proof. 

Suppose that $\mathcal{Z}=\mathfrak{F}\left(\sigma \right)$ for some $\sigma \in {\mathcal{P}}_c^{+}$. Then ∀$m_1\in (m_{-}^{{\nu }_1},m_{+}^{{\nu }_1})$ and ∀$m_2\in (m_{-}^{{\nu }_2},m_{+}^{{\nu }_2})$, there is $r\in (m_{-}^{\sigma },m_{+}^{\sigma })$, so that

$$Q_r^{\sigma }=Q_{m_1}^{{\nu }_1}⊠Q_{m_2}^{{\nu }_2}.$$

That is, ∀z close to 0

$${\mathcal{S}}_{Q_r^{\sigma }}\left(z\right)={\mathcal{S}}_{Q_{m_1}^{{\nu }_1}}\left(z\right){\mathcal{S}}_{Q_{m_2}^{{\nu }_2}}\left(z\right).$$

(12)

We know that ${\displaystyle \underset{z⟶0}{lim}{\mathcal{S}}_{\sigma }\left(z\right)=\frac{1}{m_0^{\sigma }}}$. Thus

$${\displaystyle \frac{1}{r}=\frac{1}{m_0^{Q_r^{\sigma }}}=\underset{z⟶0}{lim}{\mathcal{S}}_{Q_r^{\sigma }}\left(z\right)=\underset{z⟶0}{lim}{\mathcal{S}}_{Q_{m_1}^{{\nu }_1}}\left(z\right){\mathcal{S}}_{Q_{m_2}^{{\nu }_2}}\left(z\right)=\frac{1}{m_1{m}_2}.}$$

Then, $r=m_1{m}_2$. Using ([13], [Theorem 3.3]), the $\mathcal{S}$-transforms of measures $Q_{m_1{m}_2}^{\sigma }$, $Q_{m_1}^{{\nu }_1}$ and $Q_{m_2}^{{\nu }_2}$ may be written as

$${\mathcal{S}}_{Q_{m_1{m}_2}^{\sigma }}\left(z\right)={\displaystyle \frac{1}{m_1{m}_2}}-{\displaystyle \frac{{\mathcal{V}}_{\sigma }\left(m_1{m}_2\right)}{{\left(m_1{m}_2\right)}^3}}z++zϵ\left(z\right),\mathrm{where}ϵ\left(z\right)\stackrel{z\to 0}{\to }0$$

(13)

$${\mathcal{S}}_{Q_{m_1}^{{\nu }_1}}\left(z\right)={\displaystyle \frac{1}{m_1}}-{\displaystyle \frac{{\mathcal{V}}_{{\nu }_1}\left(m_1\right)}{m_1^3}}z+z{ϵ}_1\left(z\right),\mathrm{where}{ϵ}_1\left(z\right)\stackrel{z\to 0}{\to }0$$

(14)

and

$${\mathcal{S}}_{Q_{m_2}^{{\nu }_2}}\left(z\right)={\displaystyle \frac{1}{m_2}}-{\displaystyle \frac{{\mathcal{V}}_{{\nu }_2}\left(m_2\right)}{m_2^3}}z+z{ϵ}_2\left(z\right),\mathrm{where}{ϵ}_2\left(z\right)\stackrel{z\to 0}{\to }0.$$

(15)

Combining (12), (13), (14) and (15), we obtain

$${\displaystyle \frac{{\mathcal{V}}_{\sigma }\left(m_1{m}_2\right)}{{\left(m_1{m}_2\right)}^2}}={\displaystyle \frac{{\mathcal{V}}_{{\nu }_1}\left(m_1\right)}{m_1^2}}+{\displaystyle \frac{{\mathcal{V}}_{{\nu }_2}\left(m_2\right)}{m_2^2}}.$$

(16)

On the other hand, it is well known that the solutions the functional equation

$$f\left(x_1{x}_2\right)=f_1\left(x_1\right)+f_2\left(x_2\right)$$

(17)

where the functions $f(·)$, $f_1(·)$ and $f_2(·)$ are continuous and $x_1$, $x_2\ge 0$, is

$$f\left(x\right)=cln\left(x\right)+a_1+a_2\mathrm{and}{f}_i\left(x\right)=cln\left(x\right)+a_i,i=1,2.$$

(18)

c, $a_1$ and $a_2$ arbitrary constants.

Thus, Equation (16) implies that the VFs of $\mathfrak{F}\left({\nu }_1\right)$, $\mathfrak{F}\left({\nu }_2\right)$ and $\mathfrak{F}\left(\sigma \right)$ are given, respectively, by (9), (10) and (11). □

Remark 2.

(i) Up to affinity, we may suppose that $(m_{-}^{{\nu }_i},m_{+}^{{\nu }_i})\subset (1,+\infty )$, i = 1, 2.

(ii) 

For $c=0$, we have ${\mathcal{V}}_{{\nu }_1}\left(m\right)=a_1{m}^2$, ${\mathcal{V}}_{{\nu }_2}\left(m\right)=a_2{m}^2$ and ${\mathcal{V}}_{\sigma }\left(m\right)=(a_1+a_2)m^2$. Thus, the parameters $a_1$ and $a_2$ must strictly be positive. In that case, the measures ${\nu }_1$, ${\nu }_2$ and σ are of free Gamma type law up to affinity. For more details about free Gamma CSK family, see ([4], [Theorem 3.2]).

(iii) 

We justify the choice of the parameter c, to guarantee the positivity of the variance functions given (9), (10) and (11). That is $c\in (\kappa ,+\infty )$, where $\kappa =max\{{\displaystyle \frac{-a_1}{ln\left(m_{+}^{{\nu }_1}\right)}},{\displaystyle \frac{-a_2}{ln\left(m_{+}^{{\nu }_2}\right)}},{\displaystyle \frac{-(a_1+a_2)}{ln\left(m_{+}^{\sigma }\right)}}\}$.

2.4. Stability of CSK Families by Boolean Multiplicative Convolution Product

In [14], a multiplicative concept of Boolean additive convolution was introduced. For $\lambda \in {\mathcal{P}}_{+}$, the map

$${\Phi }_{\lambda }\left(\xi \right)={\int }_0^{+\infty }{\displaystyle \frac{x\xi }{1-x\xi }}\lambda \left(dx\right),\xi \in \mathbb{C}\setminus {\mathbb{R}}_{+}$$

is univalent in $i{\mathbb{C}}_{+}$. It is known that the circle with diameter $\left(\lambda \right(\left\{0\right\})-1,0)$ contains ${\Phi }_{\lambda }\left(i{\mathbb{C}}_{+}\right)$. In addition, $\mathbb{R}\cap {\Phi }_{\lambda }\left(i{\mathbb{C}}_{+}\right)=(\lambda \left(\left\{0\right\}\right)-1,0)$. We also have ${\Phi }_{\lambda }\left({\xi }^{-1}\right)=\xi {\mathcal{G}}_{\lambda }\left(\xi \right)-1.$ Following [15], the $\eta$-transformation of $\lambda \in {\mathcal{P}}_{+}$ is provided as follows:

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

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

The transform

$${\mathfrak{B}}_{\lambda }\left(\xi \right)={\displaystyle \frac{\xi }{{\eta }_{\lambda }\left(\xi \right)}}$$

is well defined for $\xi \in \mathbb{C}\setminus {\mathbb{R}}_{+}$. The Boolean multiplicative convolution of $\tau$ and $\lambda \in {\mathcal{P}}_{+}$ is the unique probability measure in ${\mathcal{P}}_{+}$, represented as $\lambda ⨃\tau$, that fulfills

$${\mathfrak{B}}_{\lambda ⨃\tau }\left(\xi \right)={\mathfrak{B}}_{\lambda }\left(\xi \right){\mathfrak{B}}_{\tau }\left(\xi \right),for\xi \in \mathbb{C}\setminus {\mathbb{R}}_{+}.$$

For $\tau ,\lambda \in {\mathcal{P}}_{+}$ satisfying

(i)

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

(ii)

at least one of the first moments of one of the probabilities $\lambda$ or $\tau$ is finite, the measure $\lambda ⨃\tau \in {\mathcal{P}}_{+}$ is well defined.

Theorem 4.

Let ${\nu }_1$, ${\nu }_2\in {\mathcal{P}}_c^{+}$. Introduce the family of probabilities

$$\mathcal{L}=\mathfrak{F}\left({\nu }_1\right)⨃\mathfrak{F}\left({\nu }_2\right)=\{Q_{m_1}^{{\nu }_1}⨃Q_{m_2}^{{\nu }_2},m_1\in (m_{-}^{{\nu }_1},m_{+}^{{\nu }_1})and{m}_2\in (m_{-}^{{\nu }_2},m_{+}^{{\nu }_2})\}.$$

If $\mathcal{L}$ remains a CSK family, (i.e., $\mathcal{L}=\mathfrak{F}\left(\sigma \right)$ for $\sigma \in {\mathcal{P}}_c^{+}$), then ${\nu }_1$, ${\nu }_2$ and σ are such that

$${\mathcal{V}}_{{\nu }_1}\left(m\right)=m(dln\left(m\right)+b_1),\forall m\in (m_{-}^{{\nu }_1},m_{+}^{{\nu }_1}),$$

(19)

$${\mathcal{V}}_{{\nu }_2}\left(m\right)=m(dln\left(m\right)+b_2),\forall m\in (m_{-}^{{\nu }_2},m_{+}^{{\nu }_2}),$$

(20)

and

$${\mathcal{V}}_{\sigma }\left(m\right)=m(dln\left(m\right)+b_1+b_2),\forall m\in (m_{-}^{\sigma },m_{+}^{\sigma }).$$

(21)

$b_1$ and $b_2$ are strictly positive and $d\in (\upsilon ,+\infty )$ with $\upsilon =max\{{\displaystyle \frac{-b_1}{ln\left(m_{+}^{{\nu }_1}\right)}},{\displaystyle \frac{-b_2}{ln\left(m_{+}^{{\nu }_2}\right)}},{\displaystyle \frac{-(b_1+b_2)}{ln\left(m_{+}^{\sigma }\right)}}\}$.

Proof. 

Suppose that $\mathcal{L}=\mathfrak{F}\left(\sigma \right)$ for some $\sigma \in {\mathcal{P}}_c^{+}$. Then, ∀$m_1\in (m_{-}^{{\nu }_1},m_{+}^{{\nu }_1})$ and ∀$m_2\in (m_{-}^{{\nu }_2},m_{+}^{{\nu }_2})$, there is $r\in (m_{-}^{\sigma },m_{+}^{\sigma })$, so that

$$Q_r^{\sigma }=Q_{m_1}^{{\nu }_1}⨃Q_{m_2}^{{\nu }_2}.$$

That is, ∀z close to 0

$${\mathfrak{B}}_{Q_r^{\sigma }}\left(z\right)={\mathfrak{B}}_{Q_{m_1}^{{\nu }_1}}\left(z\right){\mathfrak{B}}_{Q_{m_2}^{{\nu }_2}}\left(z\right).$$

(22)

On the other hand, the multiplicative Boolean cumulant transform is related to the additive Boolean cumulant transform as

$${\displaystyle \frac{1}{{\mathfrak{B}}_{Q_r^{\sigma }}\left(z\right)}}={\mathcal{E}}_{Q_r^{\sigma }}(1/z).$$

(23)

Based on (23), (3), (4) and (5), one can obtain

$${\displaystyle \frac{1}{{\mathfrak{B}}_{Q_r^{\sigma }}\left(z\right)}}=r+{\mathcal{V}}_{\sigma }\left(r\right)z+z\epsilon \left(z\right)\mathrm{with}\epsilon \left(z\right)\stackrel{z\to 0}{\to }0,$$

(24)

$${\displaystyle \frac{1}{{\mathfrak{B}}_{Q_{m_1}^{{\nu }_1}}\left(z\right)}}=m_1+{\mathcal{V}}_{{\nu }_1}\left(m_1\right)z+z{\epsilon }_1\left(z\right)\mathrm{with}{\epsilon }_1\left(z\right)\stackrel{z\to 0}{\to }0$$

(25)

and

$${\displaystyle \frac{1}{{\mathfrak{B}}_{Q_{m_2}^{{\nu }_2}}\left(z\right)}}=m_2+{\mathcal{V}}_{{\nu }_2}\left(m_2\right)z+z{\epsilon }_2\left(z\right)\mathrm{with}{\epsilon }_2\left(z\right)\stackrel{z\to 0}{\to }0.$$

(26)

Combining (22) with (24), (25) and (26), we obtain

$$r+{\mathcal{V}}_{\sigma }\left(r\right)z+z\epsilon \left(z\right)=m_1{m}_2+(m_1{\mathcal{V}}_{{\nu }_2}\left(m_2\right)+m_2{\mathcal{V}}_{{\nu }_1}\left(m_1\right))z+z{\epsilon }_3\left(z\right)\mathrm{with}{\epsilon }_3\left(z\right)\stackrel{z\to 0}{\to }0.$$

(27)

When $z\to 0$ in (27), we obtain $r=m_1{m}_2$. Thus, relation (27) reduces to

$${\mathcal{V}}_{\sigma }\left(m_1{m}_2\right)z+z\epsilon \left(z\right)=(m_1{\mathcal{V}}_{{\nu }_2}\left(m_2\right)+m_2{\mathcal{V}}_{{\nu }_1}\left(m_1\right))z+z{\epsilon }_3\left(z\right)\mathrm{with}{\epsilon }_3\left(z\right)\stackrel{z\to 0}{\to }0.$$

(28)

Dividing by z in both sides of (28) and let $z\to 0$, one obtains ${\displaystyle \frac{{\mathcal{V}}_{\sigma }\left(m_1{m}_2\right)}{m_1{m}_2}}={\displaystyle \frac{{\mathcal{V}}_{{\nu }_1}\left(m_1\right)}{m_1}}+{\displaystyle \frac{{\mathcal{V}}_{{\nu }_2}\left(m_2\right)}{m_2}}$. Recalling (17), we obtain from (18)

$${\displaystyle \frac{{\mathcal{V}}_{{\nu }_1}\left(m\right)}{m}}=dln\left(m\right)+b_1,$$

$${\displaystyle \frac{{\mathcal{V}}_{{\nu }_2}\left(m\right)}{m}}=dln\left(m\right)+b_2$$

and

$${\displaystyle \frac{{\mathcal{V}}_{\sigma }\left(m\right)}{m}}=dln\left(m\right)+b_1+b_2.$$

Thus, measures ${\nu }_1$, ${\nu }_2$ and $\sigma$ are characterized by their corresponding VFs provided, respectively, by (19), (20) and (21). □

Remark 3.

(i) Up to affinity, we may suppose that $(m_{-}^{{\nu }_i},m_{+}^{{\nu }_i})\subset (1,+\infty )$, i = 1, 2.

(ii) 

Note that for $d=0$, we have ${\mathcal{V}}_{{\nu }_1}\left(m\right)=b_{1}m$, ${\mathcal{V}}_{{\nu }_2}\left(m\right)=b_{2}m$ and ${\mathcal{V}}_{\sigma }\left(m\right)=(b_1+b_2)m$. Thus, the parameters $b_1$ and $b_2$ must be positive strictly. In that case the measures ${\nu }_1$, ${\nu }_2$ and σ are of Marchenko–Pastur type law up to affinity.

(iii) 

We justify the choice of the parameter υ, to guarantee the positivity of the variance functions given in (19), (20) and (21). That is $d\in (\upsilon ,+\infty )$ where $\upsilon =max\{{\displaystyle \frac{-b_1}{ln\left(m_{+}^{{\nu }_1}\right)}},{\displaystyle \frac{-b_2}{ln\left(m_{+}^{{\nu }_2}\right)}},{\displaystyle \frac{-(b_1+b_2)}{ln\left(m_{+}^{\sigma }\right)}}\}$.

3. Conclusions

This article provides significant insights into the stability of CSK families under both free and Boolean convolutions. Let $\mathfrak{F}\left({\nu }_j\right)=\{Q_{m_j}^{{\nu }_j},m_j\in (m_{-}^{{\nu }_j},m_{+}^{{\nu }_j})\}$, $j=1,2$, be two CSK families induced by ${\nu }_1$ and ${\nu }_2\in {\mathcal{P}}_c$. Introduce the family of measures

$$\mathcal{H}=\mathfrak{F}\left({\nu }_1\right)\uplus \mathfrak{F}\left({\nu }_2\right)=\{Q_{m_1}^{{\nu }_1}\uplus Q_{m_2}^{{\nu }_2},m_1\in (m_{-}^{{\nu }_1},m_{+}^{{\nu }_1})\mathrm{and}{m}_2\in (m_{-}^{{\nu }_2},m_{+}^{{\nu }_2})\}.$$

We have proved that if $\mathcal{H}$ remains a CSK family, (i.e., $\mathcal{H}=\mathfrak{F}\left(\mu \right)$ for $\mu \in {\mathcal{P}}_c$), then $\mu$, ${\nu }_1$ and ${\nu }_2$ are of the Marchenko–Pastur type law up to affinity. We obtain the same conclusion if we replace (in the definition of $\mathcal{H}$) the additive Boolean convolution ⊎ with the additive free convolution ⊞. Other cases are also studied where the additive Boolean convolution is replaced (in the definition of $\mathcal{H}$) by the multiplicative free convolution ⊠ or the multiplicative Boolean convolution ⨃. This provides to some new VFs with logarithmic terms. In fact, in many cases, the variance of the response variable increases with the mean. A logarithmic transformation can help stabilize the variance, making it more constant across different levels of the predictor variables. Furthermore, the logarithmic function has desirable mathematical properties that can simplify the analysis, such as transforming multiplicative relationships into additive ones, which are easier to handle in regression analysis. In addition, in some cases, there might be theoretical reasons based on the underlying processes generating the data that suggest a logarithmic relationship between the mean and variance.