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A dynamical characterization of Poisson-Dirichlet distributions


 
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1. Title Title of document A dynamical characterization of Poisson-Dirichlet distributions
 
2. Creator Author's name, affiliation, country Louis-Pierre Arguin; Princeton University
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Point processes, Poisson-Dirichlet distributions
 
3. Subject Subject classification 60G55; 60G57
 
4. Description Abstract We show that a slight modification of a theorem of Ruzmaikina and Aizenman on competing particle systems on the real line leads to a characterization of Poisson-Dirichlet distributions $PD(\alpha,0)$. Precisely, let $\xi$ be a proper random mass-partition i.e. a random sequence $(\xi_i,i\in N)$ such that $\xi_1 \geq \xi_2 \geq \dots \geq 0$ and $\sum_i \xi_i =1$ a.s. Consider $\{W_i\}_{i\in N}$, an iid sequence of random positive numbers whose distribution is absolutely continuous with respect to the Lebesgue measure and $E[W^\lambda]<\infty$ for all $\lambda \in R$. It is shown that, if the law of $\xi$ is invariant under the random reshuffling $$( \xi_i , i \in N) \to \left(\frac{\xi_i W_i}{\sum_j \xi_jW_j } , i \in N \right)$$ where the weights are reordered after evolution, then it must be a mixture of Poisson-Dirichlet distributions $PD(\alpha,0), \alpha\in(0,1)$.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) NSERC Canada; FQRNT Quebec; NSF
 
7. Date (YYYY-MM-DD) 2007-09-21
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1300
 
10. Identifier Digital Object Identifier 10.1214/ECP.v12-1300
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 12
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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