The Annals of Probability

Regenerative composition structures

Alexander Gnedin and Jim Pitman

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Abstract

A new class of random composition structures (the ordered analog of Kingman’s partition structures) is defined by a regenerative description of component sizes. Each regenerative composition structure is represented by a process of random sampling of points from an exponential distribution on the positive halfline, and separating the points into clusters by an independent regenerative random set. Examples are composition structures derived from residual allocation models, including one associated with the Ewens sampling formula, and composition structures derived from the zero set of a Brownian motion or Bessel process. We provide characterization results and formulas relating the distribution of the regenerative composition to the Lévy parameters of a subordinator whose range is the corresponding regenerative set. In particular, the only reversible regenerative composition structures are those associated with the interval partition of [0,1] generated by excursions of a standard Bessel bridge of dimension 2−2α for some α∈[0,1].

Article information

Source
Ann. Probab. Volume 33, Number 2 (2005), 445-479.

Dates
First available in Project Euclid: 3 March 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1109868588

Digital Object Identifier
doi:10.1214/009117904000000801

Mathematical Reviews number (MathSciNet)
MR2122798

Zentralblatt MATH identifier
1070.60034

Subjects
Primary: 60G09: Exchangeability 60C05: Combinatorial probability

Keywords
Exchangeability composition structure regenerative set sampling formula subordinator

Citation

Gnedin, Alexander; Pitman, Jim. Regenerative composition structures. Ann. Probab. 33 (2005), no. 2, 445--479. doi:10.1214/009117904000000801. https://projecteuclid.org/euclid.aop/1109868588


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References

  • Aldous, D. and Pitman, J. (1994). Brownian bridge asymptotics for random mappings. Random Structures Algorithms 5 487--512.
  • Aldous, D. and Pitman, J. (2002). Two recursive decompositions of Brownian bridge related to the asymptotics of random mappings. Technical Report 595, Dept. Statistics, Univ. California, Berkeley. Available at http://www.stat.berkeley.edu/tech-reports/.
  • Aldous, D. J. (1985). Exchangeability and related topics. École d'Été de Probabilités de Saint-Flour XIII---1983. Lecture Notes in Math. 1117 1--198. Springer, Berlin.
  • Berg, C., Christensen, J. P. R. and Ressel, P. (1984). Harmonic Analysis on Semigroups. Theory of Positive Definite and Related Functions 100. Springer, New York.
  • Bertoin, J. (1999). Subordinators: Examples and applications. École d'Été de Probabilités de Saint-Flour XXVII. Lecture Notes in Math. 1727 1--198. Springer, Berlin.
  • Bertoin, J. and Yor, M. (2001). On subordinators, self-similar Markov processes and some factorizations of the exponential variable. Electron. Comm. Probab. 6 95--106.
  • Bruss, F. T. and O'Cinneide, C. A. (1990). On the maximum and its uniqueness for geometric random samples. J. Appl. Probab. 27 598--610.
  • Carmona, P., Petit, F. and Yor, M. (1997). On the distribution and asymptotic results for exponential functionals of Lévy processes. In Exponential Functionals and Principal Values Related to Brownian Motion 73--130. Rev. Mat. Iberoamericana, Madrid.
  • Doksum, K. (1974). Tailfree and neutral random probabilities and their posterior distributions. Ann. Probab. 2 183--201.
  • Donnelly, P. and Joyce, P. (1991). Consistent ordered sampling distributions: Characterization and convergence. Adv. in Appl. Probab. 23 229--258.
  • Ewens, W. J. and Tavaré, S. (1995). The Ewens sampling formula. In Multivariate Discrete Distributions (N. S. Johnson, S. Kotz and N. Balakrishnan, eds.). Wiley, New York.
  • Feller, W. (1971). An Introduction to Probability Theory and its Applications II, 2nd ed. Wiley, New York.
  • Gnedin, A. V. (1997). The representation of composition structures. Ann. Probab. 25 1437--1450.
  • Gnedin, A. V. (1998). On the Poisson--Dirichlet limit. J. Multivariate Anal. 67 90--98.
  • Gnedin, A. V. (2004). The Bernoulli sieve. Bernoulli 10 79--96.
  • Gnedin, A. V. (2004). Three sampling formulas. Combin. Probab. Comput. 13 185--193.
  • Gnedin, A. V. and Pitman, J. (2003). Regenerative composition structures, Version 2. Available at arxiv.org/abs/math.PR/0307307v2.
  • Gnedin, A. V. and Pitman, J. (2004). Regenerative partition structures. Available at arxiv.org/abs/math.PR0408071.
  • Gnedin, A. V., Pitman, J. and Yor, M. (2003). Asymptotic laws for composition derived from transformed subordinators. Available at arxiv.org/abs/math.PR/0403438.
  • Gnedin, A. V., Pitman, J. and Yor, M. (2004). Asymptotic laws for regenarative composition: Gamma subordinators and the like. Available at arxiv.org/abs/math.PR0405440.
  • Gradinaru, M., Roynette, B., Vallois, P. and Yor, M. (1999). Abel transform and integrals of Bessel local times. Ann. Inst. H. Poincaré Probab. Statist. 35 531--572.
  • Hoppe, F. M. (1987). The sampling theory of neutral alleles and an urn model in population genetics. J. Math. Biol. 25 123--159.
  • James, L. F. (2003). Poisson calculus for spatial neutral to the right processes. Available at arxiv.org/abs/math.PR/0305053.
  • Karlin, S. (1967). Central limit theorems for certain infinite urn schemes. J. Math. Mech. 17 373--401.
  • Kerov, S. (1995). Coherent random allocations and the Ewens--Pitman formula. PDMI Preprint, Steklov Math. Institute, St. Petersburg.
  • Kingman, J. F. C. (1978). The representation of partition structures. J. London Math. Soc. 18 374--380.
  • Kingman, J. F. C. (1980). The Mathematics of Genetic Diversity. SIAM, Philadelphia, PA.
  • Maisonneuve, B. (1983). Ensembles régénératifs de la droite. Z. Wahrsch. Verw. Gebiete 63 501--510.
  • Neretin, Yu. A. (1996). The group of diffeomorphisms of a ray, and random Cantor sets. Mat. Sb. 187 73--84. [Translation in Sbornik Math. 187 857--868.]
  • Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Probab. Theory Related Fields 102 145--158.
  • Pitman, J. (1997). Partition structures derived from Brownian motion and stable subordinators. Bernoulli 3 79--96.
  • Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27 1870--1902.
  • Pitman, J. (2002). Combinatorial stochastic processes. Technical Report 621, Dept. Statistics, Univ. California, Berkeley. Available at http://www.stat.berkeley.edu/tech-reports/.
  • Pitman, J. (2003). Poisson--Kingman partitions. Science and Statistics: A Festschrift for Terry Speed 30 (D. R. Goldstein, ed.) 1--34. IMS, Hayward, CA.
  • Pitman, J. and Speed, T. P. (1973). A note on random times. Stochastic Process. Appl. 1 369--374.
  • Pitman, J. and Yor, M. (1996). Random discrete distributions derived from self-similar random sets. Electron. J. Probab. 1 1--28.
  • Pitman, J. and Yor, M. (1997). On the lengths of excursions of some Markov processes. Séminaire de Probabilités XXXI. Lecture Notes in Math. 1655 272--286. Springer, Berlin.
  • Pitman, J. and Yor, M. (1997). The two-parameter Poisson--Dirichlet distribution derived from a stable subordinator. Ann. Probab. 25 855--900.
  • Sawyer, S. and Hartl, D. (1985). A sampling theory for local selection. J. Genet. 64 21--29.
  • Young, J. E. (1995). Partition-valued stochastic processes with applications. Ph.D. dissertation, Univ. California, Berkeley.