Home | Contents | Submissions, editors, etc. | Login | Search | EJP
 Electronic Communications in Probability > Vol. 12 (2007) > Paper 28 open journal systems 

A dynamical characterization of Poisson-Dirichlet distributions

Louis-Pierre Arguin, Princeton University

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(α,0). Precisely, let ξ be a proper random mass-partition i.e. a random sequence (ξi,i∈N) such that ξ_1>= ξ_2 >= ...>= 0 and ∑i ξi=1 a.s. Consider {Wi}_{i∈N}, an iid sequence of random positive numbers whose distribution is absolutely continuous with respect to the Lebesgue measure and E[Wλ]<∞ for all λ∈ R. It is shown that, if the law of ξ is invariant under the random reshuffling
( ξi,i∈N) → ({ξi Wi}/{∑j ξj Wj},i∈ N)
where the weights are reordered after evolution, then it must be a mixture of Poisson-Dirichlet distributions PD(α,0), α∈(0,1).

Full text: PDF | PostScript

Pages: 283-290

Published on: September 21, 2007

  1. Bertoin, Jean. Random fragmentation and coagulation processes.Cambridge Studies in Advanced Mathematics, 102. Cambridge University Press, Cambridge, 2006. viii+280 pp. ISBN: 978-0-521-86728-3; 0-521-86728-2 MR2253162 (Review)
  2. Dembo, Amir; Zeitouni, Ofer. Large deviations techniques and applications.Second edition.Applications of Mathematics (New York), 38. Springer-Verlag, New York, 1998. xvi+396 pp. ISBN: 0-387-98406-2 MR1619036 (99d:60030)
  3. Diaconis, Persi; Mayer-Wolf, Eddy; Zeitouni, Ofer; Zerner, Martin P. W. The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations. Ann. Probab. 32 (2004), no. 1B, 915--938. MR2044670 (2005e:60232)
  4. Liggett, Thomas M. Random invariant measures for Markov chains, and independent particle systems. Z. Wahrsch. Verw. Gebiete 45 (1978), no. 4, 297--313. MR0511776 (80b:60135)
  5. Ruzmaikina, Anastasia; Aizenman, Michael. Characterization of invariant measures at the leading edge for competing particle systems. Ann. Probab. 33 (2005), no. 1, 82--113. MR2118860 (2005j:60105)

Support Tool
Capture Cite
View Metadata
Printer Friendly
Author Address
Email Author
Email Others

Home | Contents | Submissions, editors, etc. | Login | Search | EJP

Electronic Communications in Probability. ISSN: 1083-589X