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^[1] with jump size $\alpha$ and rate $\lambda$ arises in free probability theory as the limit of repeated free convolution

$\left( \left(1-\frac{\lambda}{N}\right)\delta_0 + \frac{\lambda}{N}\delta_\alpha\right)^{\boxplus N}$

as N → ∞.

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

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

$\mu=\begin{cases} (1-\lambda) \delta_0 + \lambda \nu,& \text{if } 0\leq \lambda \leq 1 \\ \nu, & \text{if }\lambda >1, \end{cases}$

where

$\nu = \frac{1}{2\pi\alpha t}\sqrt{4\lambda \alpha^2 - ( t - \alpha (1+\lambda))^2} \, dt$

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

This law also arises in random matrix theory as the Marchenko–Pastur law. Its free cumulants are all equal to $\lambda$.

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^[2]

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

$R(z)=\frac{\lambda \alpha}{1-\alpha z}.$

The Stieltjes transformation (also known as the Cauchy transform) is given by

$G(z) = \frac{ z + \alpha - \lambda \alpha - \sqrt{ (z-\alpha (1+\lambda))^2 - 4 \lambda \alpha^2}}{2\alpha z}$

The S-transform is given by

$S(z) = \frac{1}{z+\lambda}$

in the case that $\alpha=1$.

References

1. Free Random Variables by D. Voiculescu, K. Dykema, A. Nica, CRM Monograph Series, American Mathematical Society, Providence RI, 1992
2. Lectures on the Combinatorics of Free Probability by A. Nica and R. Speicher, pp. 203–204, Cambridge Univ. Press 2006