Free Poisson distribution - Alchetron, the free social encyclopedia

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

( ( 1 − λ N ) δ 0 + λ N δ α ) ⊞ N

as N → ∞.

In other words, let X N be random variables so that X N has value α with probability λ N and value 0 with the remaining probability. Assume also that the family X 1 , X 2 , … are freely independent. Then the limit as N → ∞ of the law of X 1 + ⋯ + X N 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

μ = { ( 1 − λ ) δ 0 + λ ν , if 0 ≤ λ ≤ 1 ν , if λ > 1 ,

where

ν = 1 2 π α t 4 λ α 2 − ( t − α ( 1 + λ ) ) 2 d t

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 all equal to λ .

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

R ( z ) = λ α 1 − α z .

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

G ( z ) = z + α − λ α − ( z − α ( 1 + λ ) ) 2 − 4 λ α 2 2 α z

The S-transform is given by

S ( z ) = 1 z + λ

in the case that α = 1 .

References

Free Poisson distribution Wikipedia

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Contents

Definition

Some transforms of this law

References