Free Poisson distribution

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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 α and rate λ arises in free probability theory as the limit of repeated free convolution

((1−λN)δ0+λNδα)⊞N

as N → ∞.

In other words, let XN be random variables so that XN has value α with probability λN and value 0 with the remaining probability. Assume also that the family X1,X2,… are freely independent. Then the limit as N->∞ of the law of X1+⋯+XN is given by the Free Poisson law with parameters λ,α.

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[2]

μ={(1−λ)δ0+ν,if 0≤λ≤1ν,if λ>1,

where

ν=12παt4λα2−(t−α(1+λ))2dt

and has support [α(1−λ)2,α(1+λ)2].

This law also arises in random matrix theory as the Marchenko-Pastur law. Its free cumulants are equal to κn=λαn.

## 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[3]

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

R(z)=λα1−αz.

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

G(z)=z+α−λα−(z−α(1+λ))2−4λα22αz

The S-transform is given by

S(z)=1z+λ

in the case that α=1.

## References

1. ↑ Free Random Variables by D. Voiculescu, K. Dykema, A. Nica, CRM Monograph Series, American Mathematical Society, Providence RI, 1992
2. ↑ James A. Mingo, Roland Speicher: Free Probability and Random Matrices. Fields Institute Monographs, Vol. 35, Springer, New York, 2017.
3. ↑ Lectures on the Combinatorics of Free Probability by A. Nica and R. Speicher, pp. 203-204, Cambridge Univ. Press 2006

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