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[edit]

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[edit]

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[edit]

  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