Free Poisson distribution
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as N → ∞.
In other words, let be random variables so that has value with probability and value 0 with the remaining probability. Assume also that the family are freely independent. Then the limit as of the law of is given by the Free Poisson law with parameters .
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
and has support .
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
The Stieltjes transformation (also known as the Cauchy transform) is given by
The S-transform is given by
in the case that .
- 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