In probability theory, Poisson-Dirichlet distributions are probability distributions on the set of nonnegative, non-increasing sequences with sum 1, depending on two parameters α ∈ [ 0 , 1 ) {\displaystyle \alpha \in [0,1)} and θ ∈ ( − α , ∞ ) {\displaystyle \theta \in (-\alpha ,\infty )} . The Poisson-Dirichlet distribution can be defined as follows. Consider independent random variables ( Y n ) n ≥ 1 {\displaystyle (Y_{n})_{n\geq 1}} such that Y n {\displaystyle Y_{n}} follows the beta distribution of parameters 1 − α {\displaystyle 1-\alpha } and θ + n α {\displaystyle \theta +n\alpha } . Then, the Poisson-Dirichlet distribution P D ( α , θ ) {\displaystyle PD(\alpha ,\theta )} of parameters α {\displaystyle \alpha } and θ {\displaystyle \theta } is the law of the random decreasing sequence containing Y 1 {\displaystyle Y_{1}} and the products Y n ∏ k = 1 n − 1 ( 1 − Y k ) {\displaystyle Y_{n}\prod _{k=1}^{n-1}(1-Y_{k})} . This definition is due to Jim Pitman and Marc Yor. It generalizes Kingman's law, which corresponds to the particular case α = 0 {\displaystyle \alpha =0} .
Applications
Pitman-Yor process The values drawn from this distribution can be used as weights for sampling an infinite sequence of discrete items, an instance of a Pitman–Yor process. This is used to model the observation process of words, or species, etc. It is useful because it can generate phenomena with heavy-tailed distributions.
Number theory Patrick Billingsley has proven the following result: if n {\displaystyle n} is a uniform random integer in { 2 , 3 , … , N } {\displaystyle \{2,3,\dots ,N\}} , if k ≥ 1 {\displaystyle k\geq 1} is a fixed integer, and if p 1 ≥ p 2 ≥ ⋯ ≥ p k {\displaystyle p_{1}\geq p_{2}\geq \dots \geq p_{k}} are the k {\displaystyle k} largest prime divisors of n {\displaystyle n} (with p j {\displaystyle p_{j}} arbitrarily defined if n {\displaystyle n} has less than j {\displaystyle j} prime factors), then the joint distribution of ( log p 1 / log n , log p 2 / log n , … , log p k / log n ) {\displaystyle (\log p_{1}/\log n,\log p_{2}/\log n,\dots ,\log p_{k}/\log n)} converges to the law of the k {\displaystyle k} first elements of a P D ( 0 , 1 ) {\displaystyle PD(0,1)} distributed random sequence, when N {\displaystyle N} goes to infinity.
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