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.


The free Poisson distribution[1] with jump size and rate arises in free probability theory as the limit of repeated free convolution

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[2]


and has support .

This law also arises in random matrix theory as the Marchenko–Pastur law. Its free cumulants are equal to .

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[3]

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

The Cauchy transform (which is the negative of the Stieltjes transformation) is given by

The S-transform is given by

in the case that .


  1. ^ Free Random Variables by D. Voiculescu, K. Dykema, A. Nica, CRM Monograph Series, American Mathematical Society, Providence RI, 1992
  2. ^ James A. Mingo, Roland Speicher: Free Probability and Random Matrices. Fields Institute Monographs, Vol. 35, Springer, New York, 2017.
  3. ^ Lectures on the Combinatorics of Free Probability by A. Nica and R. Speicher, pp. 203–204, Cambridge Univ. Press 2006