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2014-01-24T08:24:03Z

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{{expert|Probability|date=November 2009}}

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<ref>Free Random Variables by D. Voiculescu, K. Dykema, A. Nica, CRM Monograph Series, American Mathematical Society, Providence RI, 1992</ref> with jump size <math>\alpha</math> and rate <math>\lambda</math> arises in [[free probability]] theory as the limit of repeated [[free convolution]]

: <math>
\left( \left(1-\frac{\lambda}{N}\right)\delta_0 + \frac{\lambda}{N}\delta_\alpha\right)^{\boxplus N}</math>

as ''N'' → ∞.

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

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

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

where

: <math>\nu = \frac{1}{2\pi\alpha t}\sqrt{4\lambda \alpha^2 - ( t - \alpha (1+\lambda))^2} \, dt</math>

and has support <math>\alpha (1-\sqrt{\lambda})^2,\alpha (1+\sqrt{\lambda})^2]</math>.

This law also arises in [[random matrix]] theory as the [[Marchenko–Pastur law]]. Its [[free cumulants]] are all equal to <math>\lambda</math>.

==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<ref>Lectures on the Combinatorics of Free Probability by A. Nica and R. Speicher, pp. 203–204, Cambridge Univ. Press 2006</ref>

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

: <math>R(z)=\frac{\lambda \alpha}{1-\alpha z}. </math>

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

: <math>
G(z) = \frac{ z + \alpha - \lambda \alpha - \sqrt{ (z-\alpha (1+\lambda))^2 - 4 \lambda \alpha^2}}{2\alpha z}
</math>

The [[S-transform]] is given by

: <math>
S(z) = \frac{1}{z+\lambda}
</math>

in the case that <math>\alpha=1</math>.

==References==
{{Reflist}}

[[Category:Probability distributions]]
[[Category:Functional analysis]]
[[Category:Free probability theory]]

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