Free Poisson distribution

In the mathematics of free probability theory, the free Poisson distribution is a counterpart of the Poisson distribution in conventional probability theory.

Definition

The free Poisson distribution with jump size [math]\displaystyle{ \alpha }[/math] and rate [math]\displaystyle{ \lambda }[/math] arises in free probability theory as the limit of repeated free convolution

[math]\displaystyle{ \left( \left(1-\frac{\lambda}{N}\right)\delta_0 + \frac{\lambda}{N}\delta_\alpha\right)^{\boxplus N} }[/math]

as N → ∞.

In other words, let [math]\displaystyle{ X_N }[/math] be random variables so that [math]\displaystyle{ X_N }[/math] has value [math]\displaystyle{ \alpha }[/math] with probability [math]\displaystyle{ \frac{\lambda}{N} }[/math] and value 0 with the remaining probability. Assume also that the family [math]\displaystyle{ X_1,X_2,\ldots }[/math] are freely independent. Then the limit as [math]\displaystyle{ N\to\infty }[/math] of the law of [math]\displaystyle{ X_1+\cdots +X_N }[/math] is given by the Free Poisson law with parameters [math]\displaystyle{ \lambda,\alpha }[/math].

This definition is analogous to one of the ways in which the classical Poisson distribution is obtained from a (classical) Poisson process.

The measure associated to the free Poisson law is given by

[math]\displaystyle{ \mu=\begin{cases} (1-\lambda) \delta_0 + \nu,& \text{if } 0\leq \lambda \leq 1 \\ \nu, & \text{if }\lambda \gt 1, \end{cases} }[/math]

where

[math]\displaystyle{ \nu = \frac{1}{2\pi\alpha t}\sqrt{4\lambda \alpha^2 - ( t - \alpha (1+\lambda))^2} \, dt }[/math]

and has support [math]\displaystyle{ [\alpha (1-\sqrt{\lambda})^2,\alpha (1+\sqrt{\lambda})^2] }[/math].

This law also arises in random matrix theory as the Marchenko–Pastur law. Its free cumulants are equal to [math]\displaystyle{ \kappa_n=\lambda\alpha^n }[/math].

Some transforms of this law

We give values of some important transforms of the free Poisson law; the computation can be found in e.g. in the book Lectures on the Combinatorics of Free Probability by A. Nica and R. Speicher

The R-transform of the free Poisson law is given by

[math]\displaystyle{ R(z)=\frac{\lambda \alpha}{1-\alpha z}. }[/math]

The Cauchy transform (which is the negative of the Stieltjes transformation) is given by

[math]\displaystyle{ G(z) = \frac{ z + \alpha - \lambda \alpha - \sqrt{ (z-\alpha (1+\lambda))^2 - 4 \lambda \alpha^2}}{2\alpha z} }[/math]

The S-transform is given by

[math]\displaystyle{ S(z) = \frac{1}{z+\lambda} }[/math]

in the case that [math]\displaystyle{ \alpha=1 }[/math].