Hostname: page-component-74656f9884-j7cmm Total loading time: 0 Render date: 2026-11-09T10:21:40.232Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Class of Non-Regenerative Sampling Distributions

Published online by Cambridge University Press:  01 May 2007

MARTIN MÖHLE
MARTIN MÖHLE*
Affiliation:
Mathematical Institute, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany (e-mail: martin.moehle@uni-tuebingen.de)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We show that the sampling formula induced from a Λ-coalescent process with multiple collisions is regenerative if and only if the measure Λ is either concentrated in 0 (Kingman case) or concentrated in 1 (star-shaped case). The Ewens sampling formula is the only sampling formula in this class which also belongs to Pitman's two-parameter family of sampling distributions.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 16 , Issue 3 , May 2007 , pp. 435 - 444
Copyright
Copyright © Cambridge University Press 2006

References

[1]Bolthausen, E. and Sznitman, A.-S. (1998) On Ruelle's probability cascades and an abstract cavity method. Comm. Math. Phys. 197 247–276.CrossRefGoogle Scholar
[2]Donnelly, P. and Joyce, P. (1991) Consistent ordered sampling distributions: Characterization and convergence. Adv. Appl. Probab. 23 229–258.CrossRefGoogle Scholar
[3]Ewens, W. J. and Tavaré, S. (1995) The Ewens sampling formula. In Multivariate Discrete Distributions (Johnson, N. S. et al. ., eds), Wiley, New York.Google Scholar
[4]Gnedin, A. (2004) Three sampling formulas. Combin. Probab. Comput. 13 185–193.CrossRefGoogle Scholar
[5]Gnedin, A. (2004) The Bernoulli sieve. Bernoulli 10 79–96.CrossRefGoogle Scholar
[6]Gnedin, A. and Pitman, J.} (2005) Regenerative composition structures. Ann. Probab. 33 445–479.CrossRefGoogle Scholar
[7]Gnedin, A. and Pitman, J. (2005) Regenerative partition structures. Electron. J. Combin. 11 # 12.CrossRefGoogle Scholar
[8]Gnedin, A.Pitman, J. and Yor, M. (2006) Asymptotic laws for regenerative compositions: Gamma subordinators and the like. Probab. Theory Rel. Fields 135 576–602.CrossRefGoogle Scholar
[9]Gnedin, A.Pitman, J. and Yor, M. (2006) Asymptotic laws for compositions derived from transformed subordinators. Ann. Probab. 34 468–492.CrossRefGoogle Scholar
[10]Keener, R.Rothman, E. and Starr, N. (1987) Distributions on partitions. Ann. Stat. 15 1466–1481.CrossRefGoogle Scholar
[11]Kingman, J. F. C. (1982) The coalescent. Stoch. Process. Appl. 13 235–248.CrossRefGoogle Scholar
[12]Kingman, J. F. C. (2000) Origins of the coalescent: 1974–1982. Genetics 156 1461–1463.CrossRefGoogle ScholarPubMed
[13]Möhle, M. (2006) On sampling distributions for coalescent processes with simultaneous multiple collisions. Bernoulli 12 35–53.Google Scholar
[14]Pitman, J. (1995) Exchangeable and partially exchangeable random partitions. Probab. Theory Rel. Fields 102 145–158.CrossRefGoogle Scholar
[15]Pitman, J. (1999) Coalescents with multiple collisions. Ann. Probab. 27 1870–1902.CrossRefGoogle Scholar
[16]Sagitov, S. (1999) The general coalescent with asynchronous mergers of ancestral lines. J. Appl. Probab. 36 1116–1125.CrossRefGoogle Scholar