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

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

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