Free Poisson distribution
Encyclopedia
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 with jump size
and rate 
arises in free probability
theory as the limit of repeated free convolution
Free probability
Free probability is a mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue of the classical notion of independence, and it is connected with free products....
theory, the free Poisson distribution is a counterpart of the Poisson distribution
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...
in conventional probability theory.
Definition.
The free Poisson distribution with jump size
and rate arises in free probabilityFree probability
Free probability is a mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue of the classical notion of independence, and it is connected with free products....
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 independentFree independenceIn the mathematical theory of free probability, the notion of free independence was introduced by Dan Voiculescu. The definition of free independence is parallel to the classical definition of independence, except that the role of Cartesian products of measure spaces is played by the notion of a...
. 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 distributionPoisson distributionIn probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...
is obtained from a (classical) Poisson process.
The measure associated to the free Poisson law is given by
where
and has support
.
This law also arises in random matrixRandom matrixIn probability theory and mathematical physics, a random matrix is a matrix-valued random variable. Many important properties of physical systems can be represented mathematically as matrix problems...
theory as the Marchenko–Pastur law. Its free cumulants are all equal to
.
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
.
-
-
