Some Results on the Free Poisson Distribution

Abstract

Let ${\mathcal{K}}_{+}\left({\mu }_i\right)=\{Q_{s_i}^{{\mu }_i},s_i\in (m_0^{{\mu }_i},m_{+}^{{\mu }_i})\}$, $i=1,2$, be two CSK families generated by the nondegenerate probability measures ${\mu }_1$ and ${\mu }_2$ with support bounded from above. Define the set of measures $\mathcal{L}={\mathcal{K}}_{+}\left({\mu }_1\right)•{\mathcal{K}}_{+}\left({\mu }_2\right)=\{Q_{s_1}^{{\mu }_1}•Q_{s_2}^{{\mu }_2},s_1\in (m_0^{{\mu }_1},m_{+}^{{\mu }_1})\mathrm{and}{s}_2\in (m_0^{{\mu }_2},m_{+}^{{\mu }_2})\},$ where $Q_{s_1}^{{\mu }_1}•Q_{s_2}^{{\mu }_2}$ denotes the Fermi convolution of $Q_{s_1}^{{\mu }_1}$ and $Q_{s_2}^{{\mu }_2}$. We prove that if $\mathcal{L}$ is still a CSK family (that is, $\mathcal{L}={\mathcal{K}}_{+}\left(\sigma \right)$ for some nondegenerate probability measure ()$\sigma$), then the probability measures $\sigma$, ${\mu }_1$ and ${\mu }_2$ are of the free Poisson type and follow the free Poisson law up to affinity. The same result, regarding the free Poisson measure, is obtained if we consider the t-deformed free convolution $\begin{array}{|c|}\hline t\\ \hline\end{array}$ replacing the Fermi convolution • in the family of measures $\mathcal{L}$.

1. Introduction

The free Poisson measure plays, in non-commutative probability, a similar role played in classical probability by the Poisson measure. In the theory of random matrices, the free Poisson measure describes, for big rectangular random matrices, the asymptotical behavior of the singular values. However, a lot of results and properties are presented in the literature for the free Poisson measure: In free probability, the Lukacs-type property of the free Poisson measure is studied; see [1]. Furthermore, a property of the free Poisson measure is presented in [2] in relation to the concept of the Cauchy-Stieltjes Kernel (CSK) families and using the Boolean additive convolution. In addition, a short demonstration is provided in [3] for the free Poisson theorem. Further studies related to the free Poisson measure are presented in [4,5,6]. The aim of this article is to develop further properties of the free Poisson measure in relation to CSK families. Two kinds of convolutions are involved in this work, namely the t-deformed free convolution introduced in [7,8] and the Fermi convolution defined in [9]. To better present our work in this article, we need to first talk about CSK families. We also need to present some basics about the t-deformed free convolution and the Fermi convolution.

In free probability, the Cauchy-Stieltjes Kernel (CSK) families were defined recently by exploring the Cauchy–Stieltjes kernel ${(1-\vartheta y)}^{-1}$. Some results about CSK families are proved in [10] by involving compactly supported probabilities. Extended work on CSK families is given in [11] to cover probabilities with support bounded from above. Denote by ${\mathbb{P}}_{ba}$ the set of nondegenerate probabilities with support bounded from above. Let $\mu \in {\mathbb{P}}_{ba}$. With $1/{\vartheta }_{+}\left(\mu \right)=max\{0,sup\mathrm{supp}\left(\mu \right)\}$, the transform

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

(1)

converges ∀$\vartheta \in [0,{\vartheta }_{+}\left(\mu \right))$. For $\vartheta \in [0,{\vartheta }_{+}\left(\mu \right))$, we set

$${\mathbf{P}}_{(\vartheta ,\mu )}\left(dy\right)={\displaystyle \frac{1}{{\mathbf{M}}_{\mu }\left(\vartheta \right)(1-\vartheta y)}}\mu \left(dy\right).$$

The one-sided Cauchy-Stieltjes Kernel (CSK) family generated by $\mu$ is the set of probability measures

$${\mathcal{K}}_{+}\left(\mu \right)=\{{\mathbf{P}}_{(\vartheta ,\mu )}\left(dy\right):\vartheta \in (0,{\vartheta }_{+}\left(\mu \right))\}.$$

The mean function $\vartheta \mapsto {\mathbf{K}}_{\mu }\left(\vartheta \right)=\int y{\mathbf{P}}_{(\vartheta ,\mu )}\left(dy\right)$ strictly increases on $(0,{\vartheta }_{+}\left(\mu \right))$; see [11]. The interval $(m_0^{\mu },m_{+}^{\mu })={\mathbf{K}}_{\mu }\left((0,{\vartheta }_{+}\left(\mu \right))\right)$ is said to be the mean domain of ${\mathcal{K}}_{+}\left(\mu \right)$. Denote as ${\chi }_{\mu }(·)$ the reciprocal of ${\mathbf{K}}_{\mu }(·)$. For $p\in (m_0^{\mu },m_{+}^{\mu })$, denote $Q_p^{\mu }\left(dy\right)=P_{({\chi }_{\mu }\left(p\right),\mu )}\left(dy\right)$. The mean parametrization of ${\mathcal{K}}_{+}\left(\mu \right)$ is

$${\mathcal{K}}_{+}\left(\mu \right)=\{Q_p^{\mu }\left(dy\right);p\in (m_0^{\mu },m_{+}^{\mu })\}.$$

It is proven in [11] that

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

(2)

where

$$B=B\left(\mu \right)=max\{0,sup\mathrm{supp}\left(\mu \right)\}={\displaystyle \frac{1}{{\vartheta }_{+}\left(\mu \right)}},$$

(3)

and

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

(4)

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

The family will denoted as ${\mathcal{K}}_{-}\left(\mu \right)$ if $\mu$ has support bounded from below. It is parameterized by $\vartheta \in ({\vartheta }_{-}\left(\mu \right),0)$, where ${\vartheta }_{-}\left(\mu \right)$ is either $1/\mathcal{A}\left(\mu \right)$ or $-\infty$ with $\mathcal{A}=\mathcal{A}\left(\mu \right)=min\{0,infsupp(\mu \left)\right\}$. The domain of means for ${\mathcal{K}}_{-}\left(\mu \right)$ is the interval $(m_{-}^{\mu },m_0^{\mu })$ with $m_{-}^{\mu }=\mathcal{A}-1/{\mathcal{G}}_{\mu }\left(\mathcal{A}\right)$. If $\mu$ has compact support, then $\vartheta \in ({\vartheta }_{-}\left(\mu \right),{\vartheta }_{+}\left(\mu \right))$, and $\mathcal{K}\left(\mu \right)=\left\{\mu \right\}\cup {\mathcal{K}}_{+}\left(\mu \right)\cup {\mathcal{K}}_{-}\left(\mu \right)$ is a two-sided CSK family.

The map

$$p\mapsto {\mathcal{V}}_{\mu }\left(p\right)=\int {(y-p)}^2{Q}_p^{\mu }\left(dy\right)$$

(5)

is called the variance function of ${\mathcal{K}}_{+}\left(\mu \right)$; see [10]. If $m_0^{\mu }=\int y\mu \left(dy\right)$ does not exist, all elements of ${\mathcal{K}}_{+}\left(\mu \right)$ have infinite variance. The following substitute, called the pseudo-variance function ${\mathbb{V}}_{\mu }(·)$, is introduced in [11] as

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

(6)

If $m_0^{\mu }$ is finite, then ${\mathcal{V}}_{\mu }(·)$ exists, see [11], and

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

(7)

Remark 1. 

(i) 

${\mathbb{V}}_{\mu }(·)$ characterizes μ: if we set

$$\varpi =\varpi \left(p\right)=p+{\displaystyle \frac{{\mathbb{V}}_{\mu }\left(p\right)}{p}},$$

(8)

then

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

(9)

(ii) 

Consider $c\ne 0$ and $\delta \in \mathbb{R}$. Let $\phi \left(\mu \right)$ be an image of μ created by $\phi :y⟼cy+\delta$. Then, ∀p close enough to $m_0^{\phi \left(\mu \right)}=\phi \left(m_0^{\mu }\right)=c{m}_0^{\mu }+\delta$,

$${\mathbb{V}}_{\phi \left(\mu \right)}\left(p\right)={\displaystyle \frac{c^{2}p}{p-\delta }}{\mathbb{V}}_{\mu }\left({\displaystyle \frac{p-\delta }{c}}\right).$$

(10)

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

$${\mathcal{V}}_{\phi \left(\mu \right)}\left(p\right)=c^2{\mathcal{V}}_{\mu }\left({\displaystyle \frac{p-\delta }{c}}\right).$$

(11)

(iii) 

For $a\ne 0$, the free Poisson measure is

$${\displaystyle {\mathbf{FP}}^{\left(a\right)}\left(dr\right)=\frac{\sqrt{4-{(r-a)}^2}}{2\pi (1+ar)}{\mathbf{1}}_{(a-2,a+2)}\left(r\right)dr+(1-1/a^2){\delta }_{-1/a}\left(dr\right),}$$

(12)

with $m_0^{{\mathbf{FP}}^{\left(a\right)}}=0$. We have

$${\mathcal{V}}_{{\mathbf{FP}}^{\left(a\right)}}\left(p\right)=1+ap,\forall p\in (m_{-}^{{\mathbf{FP}}^{\left(a\right)}},m_{+}^{{\mathbf{FP}}^{\left(a\right)}})=(-1,1);$$

(13)

see [10], Theorem 3.2.

Next, we introduce some basics on the Fermi and t-deformed free convolutions, and we present the purpose of this paper. Denote by $\mathbb{P}$ (${\mathbb{P}}_c$ and ${\mathbb{P}}^2$, respectively) the set of real probabilities (the subsets of probabilities from $\mathbb{P}$ with compact support and with finite mean and variance, respectively). According to [9], for $\sigma \in {\mathbb{P}}^2$, the $\mathcal{B}$-transform is

$${\mathcal{B}}_{\sigma }\left(z\right)=m_0^{\sigma }z+z{E}_{{\sigma }^0}\left({\displaystyle \frac{1}{z}}\right),$$

(14)

where $m_0^{\sigma }$ is the expectation of $\sigma$, ${\sigma }^0$ is the zero expectation shift of $\sigma$ and

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

(15)

Denote by $\sigma ={\sigma }_1•{\sigma }_2$ the Fermi convolution of ${\sigma }_1,{\sigma }_2\in {\mathbb{P}}^2$. We have

$${\mathcal{B}}_{\sigma }\left(z\right)={\mathcal{B}}_{{\sigma }_1}\left(z\right)+{\mathcal{B}}_{{\sigma }_2}\left(z\right);$$

(16)

see [9], Theorem 3.1. Furthrmore, $\sigma \in {\mathbb{P}}^2$ and $m_0^{\sigma }=m_0^{{\sigma }_1}+m_0^{{\sigma }_2}$.

The t-deformation of measures is defined in [7,8] as follows: Let $\tau \in \mathbb{P}$ and $t>0$, then, by means of the Nevanlinna theorem, the map ${\mathcal{G}}_{{\tau }_t}(·)$ introduced by

$${\displaystyle \frac{1}{{\mathcal{G}}_{{\tau }_t}\left(z\right)}}={\displaystyle \frac{t}{{\mathcal{G}}_{\tau }\left(z\right)}}+(1-t)z,$$

(17)

is a Cauchy–Stieltjes transform of a measure denoted by $U_t\left(\tau \right):={\tau }_t.$ A new convolution, called the t-transformed free convolution denoted $\begin{array}{|c|}\hline t\\ \hline\end{array}$-convolution, is defined in [7,8] based on the t-transformation of measures; that is, for ${\tau }_1$ and ${\tau }_2\in \mathbb{P}$,

$${\tau }_1\begin{array}{|c|}\hline t\\ \hline\end{array}{\tau }_2=U_{1/t}(U_t\left({\tau }_1\right)⊞U_t\left({\tau }_2\right)).$$

(18)

For $\tau \in {\mathbb{P}}_c$, the ${\mathcal{R}}_{\tau }(·)$-transform of $\tau$ is produced by

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

(19)

The t-transformed free cumulant transform ${\mathcal{R}}_{\tau }^t(·)$ is

$${\mathcal{R}}_{\tau }^t\left(\xi \right):={\displaystyle \frac{1}{t}}{\mathcal{R}}_{U_t\left(\tau \right)}\left(\xi \right).$$

For ${\tau }_1$ and ${\tau }_2\in {\mathbb{P}}_c$, we have

$${\mathcal{R}}_{{\tau }_1\begin{array}{|c|}\hline t\\ \hline\end{array}{\tau }_2}^t={\mathcal{R}}_{{\tau }_1}^t\left(\xi \right)+{\mathcal{R}}_{{\tau }_2}^t\left(\xi \right).$$

(20)

The ${\mathcal{R}}_{\tau }^t(·)$-transform is a special case of the $(a,b)$-transformed free cumulant transform, defined in [12]. It corresponds to the case when $t=a=b>0$. One sees that

$${\displaystyle \underset{\xi \to 0}{lim}{\mathcal{R}}_{\tau }^t\left(\xi \right)=m_0^{\tau }.}$$

(21)

We end this section by presenting the purpose of this article. We provide some properties of the free Poisson measure in the setting of CSK families and involving the Fermi convolution and the $\begin{array}{|c|}\hline t\\ \hline\end{array}$-convolution. More precisely, for ${\mu }_1,{\mu }_2\in {\mathbb{P}}^2$, define the following set of measures:

$$\mathcal{L}={\mathcal{K}}_{+}\left({\mu }_1\right)•{\mathcal{K}}_{+}\left({\mu }_2\right)=\{Q_{y_1}^{{\mu }_1}•Q_{y_2}^{{\mu }_2},y_1\in (m_0^{{\mu }_1},m_{+}^{{\mu }_1})\mathrm{and}{y}_2\in (m_0^{{\mu }_2},m_{+}^{{\mu }_2})\}.$$

(22)

We prove that if $\mathcal{L}$ is still a CSK family, that is, $\mathcal{L}={\mathcal{K}}_{+}\left(\rho \right)$, for some nondegenerate measure $\rho$, then the probability measures $\rho$, ${\mu }_1$ and ${\mu }_2$ are of the free Poisson type and follow the free Poisson law up to affinity. The same result is obtained for the free Poisson measure (with other concepts) if we change the Fermi convolution with the $\begin{array}{|c|}\hline t\\ \hline\end{array}$-convolution in (22).

2. A Property of ${\mathbf{FP}}^{\mathbf{\left(}\mathit{a}\mathbf{\right)}}$ Based on the Fermi Convolution

Let $\mu \in {\mathbb{P}}^2$. In this section, for the results to be presented clearly, in place of the $\mathcal{B}$-transformation, we consider the $\mathcal{H}$-transformation:

$${\mathcal{H}}_{\mu }\left(\xi \right)=\xi {\mathcal{B}}_{\mu }\left({\displaystyle \frac{1}{\xi }}\right)=m_0^{\mu }+E_{{\mu }^0}\left(\xi \right)=m_0^{\mu }+\xi -{\displaystyle \frac{1}{{\mathcal{G}}_{{\mu }^0}\left(\xi \right)}}.$$

(23)

We have

$${\mathcal{H}}_{{\mu }_1•{\mu }_2}\left(\xi \right)={\mathcal{H}}_{{\mu }_1}\left(\xi \right)+{\mathcal{H}}_{{\mu }_2}\left(\xi \right).$$

(24)

Now, we provide some useful facts of the $\mathcal{H}$-transformation.

Proposition 1. 

Let $\sigma \in {\mathbb{P}}^2$ with support bounded from above.

(i) 

For $\xi \in \mathbb{C}\setminus supp\left(\sigma \right)$ such that $\xi \ne {\mathbb{V}}_{\xi }\left(x\right)/x$, the $\mathcal{H}$-transform of $Q_x^{\sigma }\in {\mathcal{K}}_{+}\left(\sigma \right)$ is

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

(25)

(ii) 

${\displaystyle \underset{w⟶+\infty }{lim}\frac{{\mathcal{H}}_{\sigma }(w+x-m_0^{\sigma })}{w}=0.}$

Proof. 

(i) Following [13] Lemma 2.3, for $w\in \mathbb{C}\setminus supp\left(\sigma \right)$ so that $w\ne x+{\mathbb{V}}_{\mu }\left(x\right)/x$, the Cauchy–Stieltjes transform of $Q_x^{\sigma }\in {\mathcal{K}}_{+}\left(\sigma \right)$ is

$${\mathcal{G}}_{Q_x^{\sigma }}\left(w\right)={\displaystyle \frac{1}{x+{\mathbb{V}}_{\sigma }\left(x\right)/x-w}}\left({\displaystyle \frac{{\mathbb{V}}_{\sigma }\left(x\right)}{x}}{\mathcal{G}}_{\sigma }\left(w\right)-1\right).$$

(26)

Combining (23) and (26), we obtain for $\xi$, such that $\xi +x\in \mathbb{C}\setminus supp\left(\sigma \right)$ and $\xi \ne {\mathbb{V}}_{\sigma }\left(x\right)/x$,

$${\mathcal{H}}_{Q_x^{\sigma }}\left(\xi \right)=\xi +x-{\displaystyle \frac{1}{{\mathcal{G}}_{Q_x^{\sigma }}(\xi +x)}}={\displaystyle \frac{(\xi +x){\mathcal{G}}_{\sigma }(\xi +x){\mathbb{V}}_{\sigma }\left(x\right)/x-(x+{\mathbb{V}}_{\sigma }\left(x\right)/x)}{{\mathcal{G}}_{\sigma }(\xi +x){\mathbb{V}}_{\sigma }\left(x\right)/x-1}}.$$

(27)

From (23), one sees that

$${\mathcal{G}}_{\sigma }(\xi +x)={\displaystyle \frac{1}{\xi +x-{\mathcal{H}}_{\sigma }(\xi +x-m_0^{\sigma })}}.$$

(28)

Combining (27) and (28), we obtain (25).

(ii) The proof follows from (23) and [2] Proposition 3.2.  □

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

Theorem 1. 

Let ${\mu }_1$, ${\mu }_2\in {\mathbb{P}}^2$ be nondegenerate with support bounded from above. Define the set of measures

$$\mathcal{L}={\mathcal{K}}_{+}\left({\mu }_1\right)•{\mathcal{K}}_{+}\left({\mu }_2\right)=\{Q_{y_1}^{{\mu }_1}•Q_{y_2}^{{\mu }_2},y_1\in (m_0^{{\mu }_1},m_{+}^{{\mu }_1})and{y}_2\in (m_0^{{\mu }_2},m_{+}^{{\mu }_2})\}.$$

(29)

If $\mathcal{L}$ is still a CSK family (that is, $\mathcal{L}={\mathcal{K}}_{+}\left(\rho \right)$ for some nondegenerate measure ρ), then the probability measures ρ, ${\mu }_1$ and ${\mu }_2$ are of the free Poisson type and follow the free Poisson law, ${\mathit{FP}}^{\left(a\right)}$, up to affinity.

Proof. 

Suppose that $\mathcal{L}={\mathcal{K}}_{+}\left(\rho \right)$ for some nondegenerate measure $\rho \in {\mathbb{P}}^2$ (without a loss of generality, we may suppose that $0\in (m_0^{\rho },m_{+}^{\rho })$). Then, $r\in (m_0^{\rho },m_{+}^{\rho })$ exists such that

$$Q_r^{\rho }=Q_{y_1}^{{\mu }_1}•Q_{y_2}^{{\mu }_2}.$$

(30)

That is, for all values of s large enough,

$${\mathcal{H}}_{Q_r^{\rho }}\left(s\right)={\mathcal{H}}_{Q_{y_1}^{{\mu }_1}}\left(s\right)+{\mathcal{H}}_{Q_{y_2}^{{\mu }_2}}\left(s\right).$$

(31)

From [14] Proposition 3 (iii), we have

$${\displaystyle \underset{w⟶+\infty }{lim}{\mathcal{H}}_{\rho }\left(w\right)=m_0^{\rho }.}$$

(32)

Then,

$${\displaystyle r=m_0^{Q_r^{\rho }}=\underset{w⟶+\infty }{lim}{\mathcal{H}}_{Q_r^{\rho }}\left(w\right)=\underset{w⟶+\infty }{lim}\left({\mathcal{H}}_{Q_{y_1}^{{\mu }_1}}\left(w\right)+{\mathcal{H}}_{Q_{y_2}^{{\mu }_2}}\left(w\right)\right)=m_0^{Q_{y_1}^{{\mu }_1}}+m_0^{Q_{y_2}^{{\mu }_2}}=y_1+y_2.}$$

(33)

Using (25) and (33), Equation (31) becomes

$$\begin{array}{c}{\displaystyle \frac{\left(y_1+y_2+\frac{{\mathbb{V}}_{\rho }(y_1+y_2)}{y_1+y_2}\right){\mathcal{H}}_{\rho }(s+y_1+y_2-m_0^{\rho })-(y_1+y_2)(s+y_1+y_2)}{\frac{{\mathbb{V}}_{\rho }(y_1+y_2)}{y_1+y_2}-(s+y_1+y_2-{\mathcal{H}}_{\rho }(s+y_1+y_2-m_0^{\rho }))}}=\hfill \\ {\displaystyle \frac{\left(y_1+\frac{{\mathbb{V}}_{{\mu }_1}\left(y_1\right)}{y_1}\right){\mathcal{H}}_{{\mu }_1}(s+y_1-m_0^{{\mu }_1})-y_1(s+y_1)}{\frac{{\mathbb{V}}_{{\mu }_1}\left(y_1\right)}{y_1}-(s+y_1-{\mathcal{H}}_{{\mu }_1}(s+y_1-m_0^{{\mu }_1}))}}+{\displaystyle \frac{\left(y_2+\frac{{\mathbb{V}}_{{\mu }_2}\left(y_2\right)}{y_2}\right){\mathcal{H}}_{{\mu }_2}(s+y_2-m_0^{{\mu }_2})-y_2(s+y_2)}{\frac{{\mathbb{V}}_{{\mu }_2}\left(y_2\right)}{y_2}-(s+y_2-{\mathcal{H}}_{{\mu }_2}(s+y_2-m_0^{{\mu }_2}))}}.\hfill \end{array}$$

(34)

The calculations of (34) give

$$\begin{array}{c}\left(y_1+y_2+{\displaystyle \frac{{\mathbb{V}}_{\rho }(y_1+y_2)}{y_1+y_2}}\right){\mathcal{H}}_{\rho }(s+y_1+y_2-m_0^{\rho }){\displaystyle \frac{{\mathbb{V}}_{{\mu }_1}\left(y_1\right)}{y_1}}{\displaystyle \frac{{\mathbb{V}}_{{\mu }_2}\left(y_2\right)}{y_2}}\hfill \\ -\left(y_1+y_2+{\displaystyle \frac{{\mathbb{V}}_{\rho }(y_1+y_2)}{y_1+y_2}}\right){\mathcal{H}}_{\rho }(s+y_1+y_2-m_0^{\rho }){\displaystyle \frac{{\mathbb{V}}_{{\mu }_1}\left(y_1\right)}{y_1}}(s+y_2-{\mathcal{H}}_{{\mu }_2}(s+y_2-m_0^{{\mu }_2}))\hfill \\ -\left(y_1+y_2+{\displaystyle \frac{{\mathbb{V}}_{\rho }(y_1+y_2)}{y_1+y_2}}\right){\mathcal{H}}_{\rho }(s+y_1+y_2-m_0^{\rho }){\displaystyle \frac{{\mathbb{V}}_{{\mu }_2}\left(y_2\right)}{y_2}}(s+y_1-{\mathcal{H}}_{{\mu }_1}(s+y_1-m_0^{{\mu }_1}))\hfill \\ +\left(y_1+y_2+{\displaystyle \frac{{\mathbb{V}}_{\rho }(y_1+y_2)}{y_1+y_2}}\right){\mathcal{H}}_{\rho }(s+y_1+y_2-m_0^{\rho })(s+y_1-{\mathcal{H}}_{{\mu }_1}(s+y_1-m_0^{{\mu }_1}))(s+y_2-{\mathcal{H}}_{{\mu }_2}(s+y_2-m_0^{{\mu }_2}))\hfill \\ -(y_1+y_2)(s+y_1+y_2){\displaystyle \frac{{\mathbb{V}}_{{\mu }_1}\left(y_1\right)}{y_1}}{\displaystyle \frac{{\mathbb{V}}_{{\mu }_2}\left(y_2\right)}{y_2}}+(y_1+y_2)(s+y_1+y_2){\displaystyle \frac{{\mathbb{V}}_{{\mu }_1}\left(y_1\right)}{y_1}}(s+y_2-{\mathcal{H}}_{{\mu }_2}(s+y_2-m_0^{{\mu }_2}))\hfill \\ +(y_1+y_2)(s+y_1+y_2){\displaystyle \frac{{\mathbb{V}}_{{\mu }_2}\left(y_2\right)}{y_2}}(s+y_1-{\mathcal{H}}_{{\mu }_1}(s+y_1-m_0^{{\mu }_1}))\hfill \\ +(y_1+y_2)(s+y_1+y_2)(s+y_1){\mathcal{H}}_{{\mu }_2}(s+y_2-m_0^{{\mu }_2})+(y_1+y_2)(s+y_1+y_2)(s+y_2){\mathcal{H}}_{{\mu }_1}(s+y_1-m_0^{{\mu }_1})\hfill \\ -(y_1+y_2)(s+y_1+y_2){\mathcal{H}}_{{\mu }_1}(s+y_1-m_0^{{\mu }_1}){\mathcal{H}}_{{\mu }_2}(s+y_2-m_0^{{\mu }_2})\hfill \\ ={\displaystyle \frac{{\mathbb{V}}_{\rho }(y_1+y_2)}{y_1+y_2}}\left(y_1+{\displaystyle \frac{{\mathbb{V}}_{{\mu }_1}\left(y_1\right)}{y_1}}\right){\mathcal{H}}_{{\mu }_1}(s+y_1-m_0^{{\mu }_1}){\displaystyle \frac{{\mathbb{V}}_{{\mu }_2}\left(y_2\right)}{y_2}}\hfill \\ -{\displaystyle \frac{{\mathbb{V}}_{\rho }(y_1+y_2)}{y_1+y_2}}\left(y_1+{\displaystyle \frac{{\mathbb{V}}_{{\mu }_1}\left(y_1\right)}{y_1}}\right){\mathcal{H}}_{{\mu }_1}(s+y_1-m_0^{{\mu }_1})(s+y_2-{\mathcal{H}}_{{\mu }_2}(s+y_2-m_0^{{\mu }_2}))\hfill \\ -{\displaystyle \frac{{\mathbb{V}}_{\rho }(y_1+y_2)}{y_1+y_2}}{y}_1(s+y_1){\displaystyle \frac{{\mathbb{V}}_{{\mu }_2}\left(y_2\right)}{y_2}}+{\displaystyle \frac{{\mathbb{V}}_{\rho }(y_1+y_2)}{y_1+y_2}}{y}_1(s+y_1)(s+y_2-{\mathcal{H}}_{{\mu }_2}(s+y_2-m_0^{{\mu }_2}))\hfill \\ +{\displaystyle \frac{{\mathbb{V}}_{\rho }(y_1+y_2)}{y_1+y_2}}\left(y_2+{\displaystyle \frac{{\mathbb{V}}_{{\mu }_2}\left(y_2\right)}{y_2}}\right){\mathcal{H}}_{{\mu }_2}(s+y_2-m_0^{{\mu }_2}){\displaystyle \frac{{\mathbb{V}}_{{\mu }_1}\left(y_1\right)}{y_1}}\hfill \\ -{\displaystyle \frac{{\mathbb{V}}_{\rho }(y_1+y_2)}{y_1+y_2}}\left(y_2+{\displaystyle \frac{{\mathbb{V}}_{{\mu }_2}\left(y_2\right)}{y_2}}\right){\mathcal{H}}_{{\mu }_2}(s+y_2-m_0^{{\mu }_2})(s+y_1-{\mathcal{H}}_{{\mu }_1}(s+y_1-m_0^{{\mu }_1}))\hfill \\ -{\displaystyle \frac{{\mathbb{V}}_{\rho }(y_1+y_2)}{y_1+y_2}}{y}_2(s+y_2){\displaystyle \frac{{\mathbb{V}}_{{\mu }_1}\left(y_1\right)}{y_1}}+{\displaystyle \frac{{\mathbb{V}}_{\rho }(y_1+y_2)}{y_1+y_2}}{y}_2(s+y_2)(s+y_1-{\mathcal{H}}_{{\mu }_1}(s+y_1-m_0^{{\mu }_1}))\hfill \\ -(s+y_1+y_2-{\mathcal{H}}_{\rho }(s+y_1+y_2-m_0^{\rho }))\left(y_1+{\displaystyle \frac{{\mathbb{V}}_{{\mu }_1}\left(y_1\right)}{y_1}}\right){\mathcal{H}}_{{\mu }_1}(s+y_1-m_0^{{\mu }_1}){\displaystyle \frac{{\mathbb{V}}_{{\mu }_2}\left(y_2\right)}{y_2}}\hfill \\ +(s+y_1+y_2-{\mathcal{H}}_{\rho }(s+y_1+y_2-m_0^{\rho }))\left(y_1+{\displaystyle \frac{{\mathbb{V}}_{{\mu }_1}\left(y_1\right)}{y_1}}\right){\mathcal{H}}_{{\mu }_1}(s+y_1-m_0^{{\mu }_1})(s+y_2-{\mathcal{H}}_{{\mu }_2}(s+y_2-m_0^{{\mu }_2}))\hfill \\ +(s+y_1+y_2-{\mathcal{H}}_{\rho }(s+y_1+y_2-m_0^{\rho }))y_1(s+y_1){\displaystyle \frac{{\mathbb{V}}_{{\mu }_2}\left(y_2\right)}{y_2}}\hfill \\ +(s+y_1+y_2)y_1(w+y_1){\mathcal{H}}_{{\mu }_2}(s+y_2-m_0^{{\mu }_2})+{\mathcal{H}}_{\rho }(s+y_1+y_2-m_0^{\rho })y_1(w+y_1)(w+y_2)\hfill \\ -{\mathcal{H}}_{\rho }(s+y_1+y_2-m_0^{\rho })y_1(s+y_1){\mathcal{H}}_{{\mu }_2}(s+y_2-m_0^{{\mu }_2})-{\mathcal{H}}_{\rho }(s+y_1+y_2-m_0^{\rho })y_2(s+y_2){\mathcal{H}}_{{\mu }_1}(s+y_1-m_0^{{\mu }_1})\hfill \\ +(s+y_1+y_2)y_2(s+y_2){\mathcal{H}}_{{\mu }_1}(s+y_1-m_0^{{\mu }_1})+{\mathcal{H}}_{\rho }(s+y_1+y_2-m_0^{\rho })y_2(s+y_2)(s+y_1)\hfill \\ -(s+y_1+y_2-{\mathcal{H}}_{\rho }(s+y_1+y_2-m_0^{\rho }))\left(y_2+{\displaystyle \frac{{\mathbb{V}}_{{\mu }_2}\left(y_2\right)}{y_2}}\right){\mathcal{H}}_{{\mu }_2}(s+y_2-m_0^{{\mu }_2}){\displaystyle \frac{{\mathbb{V}}_{{\mu }_1}\left(y_1\right)}{y_1}}\hfill \\ +(s+y_1+y_2-{\mathcal{H}}_{\rho }(s+y_1+y_2-m_0^{\rho }))\left(y_2+{\displaystyle \frac{{\mathbb{V}}_{{\mu }_2}\left(y_2\right)}{y_2}}\right){\mathcal{H}}_{{\mu }_2}(s+y_2-m_0^{{\mu }_2})(s+y_1-{\mathcal{H}}_{{\mu }_1}(s+y_1-m_0^{{\mu }_1}))\hfill \\ +(s+y_1+y_2-{\mathcal{H}}_{\rho }(s+y_1+y_2-m_0^{\rho }))y_2(s+y_2){\displaystyle \frac{{\mathbb{V}}_{{\mu }_1}\left(y_1\right)}{y_1}}.\hfill \end{array}$$

(35)

Divide by $s^2$ on both sides of (35), and let $s\to +\infty$. Using (32) and Proposition 1 (ii), we obtain

$${\displaystyle \frac{{\mathbb{V}}_{\rho }(y_1+y_2)}{y_1+y_2}}[y_1+y_2-m_0^{\rho }]={\displaystyle \frac{{\mathbb{V}}_{{\mu }_1}\left(y_1\right)}{y_1}}[y_1-m_0^{{\mu }_1}]+{\displaystyle \frac{{\mathbb{V}}_{{\mu }_2}\left(y_2\right)}{y_2}}[y_2-m_0^{{\mu }_2}].$$

(36)

Relation (7) together with (36) give

$${\mathcal{V}}_{\rho }(y_1+y_2)={\mathcal{V}}_{{\mu }_1}\left(y_1\right)+{\mathcal{V}}_{{\mu }_2}\left(y_2\right).$$

(37)

According to [15], relation (37) implies that

$${\mathcal{V}}_{\rho }\left(m\right)={\mathcal{V}}_{{\mu }_1}\left(m\right)=\lambda m+\beta \mathrm{and}{\mathcal{V}}_{{\mu }_2}\left(m\right)=\lambda m\mathrm{for}\lambda \ne 0\mathrm{and}\beta >0.$$

(38)

Note that the parameter $\beta ={\mathcal{V}}_{\rho }\left(0\right)$ must strictly be positive. Furthermore, $m_0^{{\mu }_2}\ne 0$; otherwise, $m\mapsto {\mathcal{V}}_{{\mu }_2}\left(m\right)=\lambda m$ cannot serve as a variance function; see [2], page 6. It is clear from (38) and (13) that $\rho$, ${\mu }_1$ and ${\mu }_2$ are of the free Poisson type and follow the free Poisson law up to affinity.  □

3. A Property of ${\mathbf{FP}}^{\mathbf{\left(}\mathit{a}\mathbf{\right)}}$ Based on the $\begin{array}{|c|}\hline \mathit{t}\\ \hline\end{array}$-Convolution

In this section, we prove that the $\begin{array}{|c|}\hline t\\ \hline\end{array}$-convolution of two CSK families is still a CSK family only in the case of a free Poisson measure. More precisely, we have the following:

Theorem 2. 

Let ${\rho }_1$, ${\rho }_2\in {\mathbb{P}}_c$ be nondegenerate. Define the set of measures

$$\mathcal{T}=\mathcal{K}\left({\rho }_1\right)\begin{array}{|c|}\hline t\\ \hline\end{array}\mathcal{K}\left({\rho }_2\right)=\{Q_{y_1}^{{\rho }_1}\begin{array}{|c|}\hline t\\ \hline\end{array}{Q}_{y_2}^{{\rho }_2},y_1\in (m_{-}^{{\rho }_1},m_{+}^{{\rho }_1})and{y}_2\in (m_{-}^{{\rho }_2},m_{+}^{{\rho }_2})\}.$$

(39)

If $\mathcal{T}$ is still a CSK family, that is, $\mathcal{T}=\mathcal{K}\left(\omega \right)$ for $\omega \in {\mathbb{P}}_c$, then the measures ω, ${\rho }_1$ and ${\rho }_2$ are of the free Poisson type and follow the law ${\mathit{FP}}^{\left(a\right)}$ up to affinity.

Proof. 

Suppose that $\mathcal{T}=\mathcal{K}\left(\omega \right)$ for some $\omega \in {\mathbb{P}}_c$. (Without a loss of generality, we may suppose that $0\in (m_{-}^{\omega },m_{+}^{\omega })$). Then, $r\in (m_{-}^{\omega },m_{+}^{\omega })$ exists such that

$$Q_r^{\omega }=Q_{y_1}^{{\rho }_1}\begin{array}{|c|}\hline t\\ \hline\end{array}{Q}_{y_2}^{{\rho }_2}.$$

(40)

That is, for ∀$\xi$ in the neighborhood of 0,

$${\mathcal{R}}_{Q_r^{\omega }}^t\left(\xi \right)={\mathcal{R}}_{Q_{y_1}^{{\rho }_1}}^t\left(\xi \right)+{\mathcal{R}}_{Q_{y_2}^{{\rho }_2}}^t\left(\xi \right).$$

(41)

Using (21) and (41), we obtain

$$\begin{array}{c}\hfill {\displaystyle r=m_0^{Q_r^{\omega }}=\underset{\xi ⟶0}{lim}{\mathcal{R}}_{Q_r^{\omega }}^t\left(\xi \right)=\underset{\xi ⟶0}{lim}\left({\mathcal{R}}_{Q_{y_1}^{{\rho }_1}}^t\left(\xi \right)+{\mathcal{R}}_{Q_{y_2}^{{\rho }_2}}^t\left(\xi \right)\right)=y_1+y_2.}\end{array}$$

(42)

The free cumulant transform of $Q_r^{\omega }$ may be written as

$${\mathcal{R}}_{Q_r^{\omega }}\left(\xi \right)=c_1\left(Q_r^{\omega }\right)+c_2\left(Q_r^{\omega }\right)\xi +\xi ϵ\left(\xi \right),\forall \xi \mathrm{in} \mathrm{the} \mathrm{neighborhood} \mathrm{of} 0.$$

(43)

where $c_1\left(Q_r^w\right)$ and $c_2\left(Q_r^{\omega }\right)$ are the free cumulants of $Q_r^{\omega }$ of order 1 and 2 respectively and ${lim}_{\xi \to 0}ϵ\left(z\right)=0$. Then

$${\mathcal{R}}_{Q_r^{\omega }}\left(\xi \right)=r+{\mathcal{V}}_{\omega }\left(r\right)\xi +\xi ϵ\left(\xi \right),\forall \xi \mathrm{in} \mathrm{a} \mathrm{neighborhood} \mathrm{of}0.$$

(44)

Using (43), the ${\mathcal{R}}^t$-transformation of $Q_m^{\mu }$ is

$$\begin{array}{c}{\mathcal{R}}_{Q_r^{\omega }}^t\left(\xi \right)={\displaystyle \frac{1}{t}}{\mathcal{R}}_{U_t\left(Q_r^{\omega }\right)}\left(\xi \right)={\displaystyle \frac{1}{t}}[c_1\left(U_t\left(Q_r^{\omega }\right)\right)+c_2\left(U_t\left(Q_r^{\omega }\right)\right)\xi +\xi ϵ\left(\xi \right)]\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{1}{t}}[tr+t{\mathcal{V}}_{\omega }\left(r\right)\xi +\xi ϵ\left(\xi \right)]=r+{\mathcal{V}}_{\omega }\left(r\right)\xi +{\displaystyle \frac{\xi }{t}}ϵ\left(\xi \right).\end{array}$$

(45)

We also have

$${\mathcal{R}}_{Q_{y_1}^{{\rho }_1}}^t\left(\xi \right)=y_1+{\mathcal{V}}_{{\rho }_1}\left(y_1\right)\xi +{\displaystyle \frac{\xi }{t}}{\epsilon }_1\left(\xi \right),\mathrm{where}{\epsilon }_1\left(\xi \right)\stackrel{\xi \to 0}{\to }0.$$

(46)

$${\mathcal{R}}_{Q_{y_2}^{{\rho }_2}}^t\left(\xi \right)=y_2+{\mathcal{V}}_{{\rho }_2}\left(y_2\right)\xi +{\displaystyle \frac{\xi }{t}}{\epsilon }_2\left(\xi \right),\mathrm{where}{\epsilon }_2\left(\xi \right)\stackrel{\xi \to 0}{\to }0.$$

(47)

Combining (41), (42), (45), (46) and (47), we obtain

$${\mathcal{V}}_{\omega }(y_1+y_2)={\mathcal{V}}_{{\rho }_1}\left(y_1\right)+{\mathcal{V}}_{{\rho }_2}\left(y_2\right).$$

(48)

According to [15], relation (48) gives

$${\mathcal{V}}_{\omega }\left(m\right)={\mathcal{V}}_{{\rho }_1}\left(m\right)=\lambda m+\beta \mathrm{and}{\mathcal{V}}_{{\rho }_2}\left(m\right)=\lambda m\mathrm{for}\lambda \ne 0\mathrm{and}\beta >0.$$

(49)

Note that $\beta ={\mathcal{V}}_{\omega }\left(0\right)$ must be strictly positive. Furthermore, $m_0^{{\rho }_2}\ne 0$; otherwise, $m\mapsto {\mathcal{V}}_{{\rho }_2}\left(m\right)=\lambda m$ cannot serve as a variance function; see [2], page 6. It is clear from (49) and (13) that $\omega$, ${\rho }_1$ and ${\rho }_2$ follow the free Poisson law up to affinity.  □

4. Conclusions

In this paper, we have explored two types of convolutions of importance in free probability: the Fermi and the t-transformed free convolutions. For ${\rho }_1$ and ${\rho }_2\in {\mathbb{P}}_c$, we define the set

$$\mathcal{T}=\mathcal{K}\left({\rho }_1\right)\begin{array}{|c|}\hline t\\ \hline\end{array}\mathcal{K}\left({\rho }_2\right)=\{Q_{y_1}^{{\rho }_1}\begin{array}{|c|}\hline t\\ \hline\end{array}{Q}_{y_2}^{{\rho }_2},y_1\in (m_{-}^{{\rho }_1},m_{+}^{{\rho }_1})\mathrm{and}{y}_2\in (m_{-}^{{\rho }_2},m_{+}^{{\rho }_2})\}.$$

(50)

We have showed that if the family $\mathcal{T}$ is a CSK family (that is, $\mathcal{T}=\mathcal{K}\left(\omega \right)$ for $\omega \in {\mathbb{P}}_c$), then the measures $\omega$, ${\rho }_1$ and ${\rho }_2$ follow the free Poisson law. The demonstration is based on characteristics of the t-transformed free cumulant transform, and an important role here is played by the variance function. An analogous property related to the free Poisson measure is proved with other tools by taking the Fermi convolution in place of the $\begin{array}{|c|}\hline t\\ \hline\end{array}$-convolution.