Free Poisson distribution

In free probability theory , the free Poisson distribution is the counterpart to the Poisson distribution from common probability theory.

definition

The free Poisson distribution with parameters and results in the free probability theory as the limit value of the iterated free convolution ${\displaystyle \alpha }$ ${\displaystyle \lambda }$

${\displaystyle \left(\left(1-{\frac {\lambda }{N}}\right)\delta _{0}+{\frac {\lambda }{N}}\delta _{\alpha }\right)^{\boxplus N}}$

for . ${\displaystyle N\to \infty }$

More precisely: be a random variable , so that the value with probability and the value 0 with probability . Continue to be the family freely independent in the sense of free probability theory . Then the distribution of in the limit value is given by a free Poisson distribution with the parameters and . ${\displaystyle X_{N}}$ ${\displaystyle X_{N}}$ ${\displaystyle \alpha }$ ${\displaystyle \lambda /N}$ ${\displaystyle 1-\lambda /N}$ ${\displaystyle X_{1},X_{2},\ldots }$ ${\displaystyle X_{1}+\cdots +X_{N}}$ ${\displaystyle N\to \infty }$ ${\displaystyle \alpha }$ ${\displaystyle \lambda }$

This definition is analogous to a corresponding limit theorem for the classic Poisson distribution with regard to the classic convolution .

Explicit form

Explicitly, the free Poisson distribution has the following form

${\displaystyle \mu ={\begin{cases}(1-\lambda )\delta _{0}+\nu ,&{\text{wenn }}0\leq \lambda \leq 1\\\nu ,&{\text{wenn }}\lambda >1,\end{cases}}}$

in which

${\displaystyle \nu ={\frac {1}{2\pi \alpha t}}{\sqrt {4\lambda \alpha ^{2}-(t-\alpha (1+\lambda ))^{2}}}\,dt}$

has the carrier . Your free accumulators are given by . ${\displaystyle [\alpha (1-{\sqrt {\lambda }})^{2},\alpha (1+{\sqrt {\lambda }})^{2}]}$ ${\displaystyle \kappa _{n}=\lambda \alpha ^{n}}$

Relation to random matrices

The free Poisson distribution appears in the theory of random matrices as the Marchenko-Pastur distribution .

Individual evidence

1. ↑ Lectures on the Combinatorics of Free Probability by A. Nica and R. Speicher, pp. 203-204, Cambridge Univ. Press 2006
2. ↑ James A. Mingo, Roland Speicher: Free Probability and Random Matrices. Fields Institute Monographs, Vol. 35, Springer, New York, 2017.