Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-16T16:17:24.520Z 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: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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