In probability theory, Poisson-Dirichlet distributions are probability distributions on the set of nonnegative, non-increasing sequences with sum 1, depending on two parameters
θ ∈ ( − α , ∞ )
One considers independent random variables
follows the beta distribution of parameters
θ + n α
Then, the Poisson-Dirichlet distribution
( α , θ )
is the law of the random decreasing sequence containing
This definition is due to Jim Pitman and Marc Yor.
[1][2] It generalizes Kingman's law, which corresponds to the particular case
[3] Patrick Billingsley[4] has proven the following result: if
is a uniform random integer in
is a fixed integer, and if
largest prime divisors of
arbitrarily defined if
prime factors), then the joint distribution of
converges to the law of the
distributed random sequence, when
The Poisson-Dirichlet distribution of parameters
θ = 1
is also the limiting distribution, for
going to infinity, of the sequence
largest cycle of a uniformly distributed permutation of order
θ > 0
, one replaces the uniform distribution by the distribution
, θ
is the number of cycles of the permutation
, then we get the Poisson-Dirichlet distribution of parameters
The probability distribution
is called Ewens's distribution,[5] and comes from the Ewens's sampling formula, first introduced by Warren Ewens in population genetics, in order to describe the probabilities associated with counts of how many different alleles are observed a given number of times in the sample.