Poisson-Dirichlet distribution

In probability theory, Poisson-Dirichlet distributions are probability distributions on the set of nonnegative, non-increasing sequences with sum 1, depending on two parameters and . It can be defined as follows. One considers independent random variables such that follows the beta distribution of parameters and . Then, the Poisson-Dirichlet distribution of parameters and is the law of the random decreasing sequence containing and the products . This definition is due to Jim Pitman and Marc Yor.[1][2] It generalizes Kingman's law, which corresponds to the particular case .[3]

Number theory

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Patrick Billingsley[4] has proven the following result: if   is a uniform random integer in  , if   is a fixed integer, and if   are the   largest prime divisors of   (with   arbitrarily defined if   has less than   prime factors), then the joint distribution of converges to the law of the   first elements of a   distributed random sequence, when   goes to infinity.

Random permutations and Ewens's sampling formula

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The Poisson-Dirichlet distribution of parameters   and   is also the limiting distribution, for   going to infinity, of the sequence  , where   is the length of the   largest cycle of a uniformly distributed permutation of order  . If for  , one replaces the uniform distribution by the distribution   on   such that  , where   is the number of cycles of the permutation  , then we get the Poisson-Dirichlet distribution of parameters   and  . 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.

References

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  1. ^ Pitman, Jim; Yor, Marc (1997). "The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator". Annals of Probability. 25 (2): 855–900. CiteSeerX 10.1.1.69.1273. doi:10.1214/aop/1024404422. MR 1434129. Zbl 0880.60076.
  2. ^ Bourgade, Paul. "Lois de Poisson–Dirichlet". Master thesis.
  3. ^ Kingman, J. F. C. (1975). "Random discrete distributions". J. Roy. Statist. Soc. Ser. B. 37: 1–22.
  4. ^ Billingsley, P. (1972). "On the distribution of large prime divisors". Periodica Mathematica. 2: 283–289.
  5. ^ Ewens, Warren (1972). "The sampling theory of selectively neutral alleles". Theoretical Population Biology. 3: 87–112.