## Coalescents With Multiple Collisions (1999)

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Venue: | Ann. Probab |

Citations: | 116 - 11 self |

### BibTeX

@ARTICLE{Pitman99coalescentswith,

author = {Jim Pitman},

title = {Coalescents With Multiple Collisions},

journal = {Ann. Probab},

year = {1999},

volume = {27},

pages = {1870--1902}

}

### Years of Citing Articles

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### Abstract

For each finite measure on [0

### Citations

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716 |
A Bayesian analysis of some nonparametric problems
- Ferguson
- 1973
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Citation Context ...rd exponential distribution. To restate (i), the random measure on (0; 1) which assigns mass ~ f 1 (t) to [0; t] is a Dirichlet random measure governed by the standard exponential distribution, as in =-=[19]-=-. Parts (ii) and (iii) are equivalents of (i) by well known properties of the Dirichlet random measure [12, Theorem 3.1],[19]. To interpret these results, regard the frequencies of blocks of \Pi 1 (t)... |

221 | The two parameter Poisson-Dirichlet distribution derived from a stable subordinator
- Pitman, Yor
- 1997
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Citation Context ...rtition of N characterized by the EPF (15), or by frequencies of the form (12)-(13), an (ff; `) partition. It was shown in [31] how to construct an (ff; `) partition by a simple urn scheme. Following =-=[37]-=-, define the Poisson-Dirichlet distribution with parameters (ff; `), abbreviated PD(ff; `), to be the distribution of ranked frequencies of an (ff; `) partition. That is, PD(ff; `) is the distribution... |

214 |
The coalescent
- Kingman
- 1982
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Citation Context ...e restriction ofsto [n]. Give P1 the topology it inherits as a subset of P 1 \Theta P 2 \Theta \Delta \Delta \Delta with the product of discrete topologies. So P1 is compact and metrizable. Following =-=[25, 15]-=-, call a P1 -valued stochastic process \Pi 1 := (\Pi 1 (t); ts0) a coalescent if \Pi 1 has c`adl`ag paths and \Pi 1 (s) a refinement of \Pi 1 (t) for every s ! t. That is to say, for each n the restri... |

142 | Deterministic and stochastic model for coalescence (aggregation and coagulation): A review of the mean-field theory and probabilists
- Aldous
- 1999
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Citation Context ...DMS97-03961 1 1 Introduction Markovian coalescent models for the evolution of a system of masses by a random process of binary collisions were introduced by Marcus [29] and Lushnikov [28]. See Aldous =-=[3]-=- for a recent survey of the scientific literature of these models and their relation to Smoluchowski's mean-field theory of coagulation phenomena. Evans and Pitman [15] gave a general framework for th... |

140 |
On the genealogy of large populations
- Kingman
- 1982
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Citation Context ... partition of N into singletons. Call a-coalescent started in state 1 1 a standard -coalescent. Fors= ffi 0 , the transition rates aresb;k = 1(k = 2). So the ffi 0 -coalescent is Kingman's coalescent =-=[25, 27]-=- in which each pair of blocks coalesces at rate 1, and no multiple collisions are allowed. For r; s ? 0 ands= beta(r; s), the probability distribution on (0; 1) with density B(r; s) \Gamma1 x r\Gamma1... |

127 |
Exchangeability and related topics
- Aldous
- 1985
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Citation Context ...ch Y 1 ; Y 2 ; : : : are conditionally independent with distribution G given some random probability distribution G. The corresponding P is then the distribution of ranked sizes of atoms of G. Aldous =-=[2]-=- gave a quick proof of Kingman's correspondence based on de Finetti's theorem. If the sequence of ranked frequencies f = (f 1 ; f 2 ; : : :) of the exchangeable random partition \Pi is proper, meaning... |

106 |
Exchangeable and partially exchangeable random partitions
- Pitman
- 1995
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Citation Context ...ection 2.3 presents a theorem which shows how certain operations of coagulation and fragmentation act on the two-parameter family of distributions of exchangeable random partitions of N introduced in =-=[31]-=- and studied further in [33, 34]. Section 2.4 applies this theorem to the U-coalescent to recover some of the results of Bolthausen-Sznitman and to obtain various further developments. The conceptual ... |

97 | Continuous multivariate distributions - Johnson, Kotz - 1972 |

88 |
Random discrete distributions
- Kingman
(Show Context)
Citation Context ...nking the jumps of a subordinator ( t ; 0sts1), that is an increasing process with stationary independent increments, with E exp(\Gamma s ) = exp ` \Gammas Z 1 0 (1 \Gamma e \Gammax )(dx) ' fors0: (i)=-=[22, 23]-=- If (dx) = `x \Gamma1 e \Gammax dx for ` ? 0, corresponding tos1 with the gamma(`) distribution P( 1 2 dx) = \Gamma(`) \Gamma1 x `\Gamma1 e \Gammax dx, then V has PD(0; `) distribution. (ii)[30] If (d... |

87 | Some developments of the Blackwell-MacQueen urn scheme
- Pitman
- 1996
(Show Context)
Citation Context ...m which shows how certain operations of coagulation and fragmentation act on the two-parameter family of distributions of exchangeable random partitions of N introduced in [31] and studied further in =-=[33, 34]-=-. Section 2.4 applies this theorem to the U-coalescent to recover some of the results of Bolthausen-Sznitman and to obtain various further developments. The conceptual framework of the paper is provid... |

84 | Brownian excursions, critical random graphs and the multiplicative coalescent
- Aldous
- 1997
(Show Context)
Citation Context ...ISIONS Jim Pitman Technical Report No. 495 Department of Statistics, University of California 367 Evans Hall # 3860, Berkeley, CA 94720-3860 Revised March 23, 1999 Abstract For each finite measureson =-=[0; 1]-=-, a coalescent Markov process, with state space the compact set of all partitions of the set N of positive integers, is constructed so the restriction of the partition to each finite subset of N is a ... |

84 |
The sampling theory of selectively neutral alleles
- Ewens
- 1972
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Citation Context ...as well known applications in population genetics, number theory, and combinatorics, as reviewed in [17, 6]. Formula (15) in this case is a variation due to Kingman [26] of the Ewens sampling formula =-=[16]-=-, [20, Ch. 41]. See [30, 37] for interpretations of PD(ff; 0) in terms of excursions of a Markov process such as a Brownian motion or a recurrent Bessel process whose zero set is the closed range of a... |

74 |
On Ruelle’s probability cascades and an abstract cavity method
- Bolthausen, Sznitman
(Show Context)
Citation Context ...0 , a unit mass at 0, is Kingman's coalescent in which every pair of blocks coalesces at rate 1. The cases= U , the uniform distribution on [0; 1] yields the coalescent derived by Bolthausen-Sznitman =-=[9]-=- from Ruelle's probability cascades [39]. See also [8] for another derivation of this coalescent from the genealogy of a continuous-state branching process. The rest of this paper is organized as foll... |

66 |
Size-biased sampling of Poisson point processes and excursions
- Perman, Pitman, et al.
- 1992
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Citation Context ...i)[22, 23] If (dx) = `x \Gamma1 e \Gammax dx for ` ? 0, corresponding tos1 with the gamma(`) distribution P( 1 2 dx) = \Gamma(`) \Gamma1 x `\Gamma1 e \Gammax dx, then V has PD(0; `) distribution. (ii)=-=[30]-=- If (dx) = cx \Gammaff\Gamma1 dx for ff 2 (0; 1) and c ? 0, corresponding tos1 with a stable distribution of index ff, then V has PD(ff; 0) distribution. The PD(0; `) distribution has well known appli... |

66 | Lines-of-descent and genealogical processes, and their application in population genetic models - Tavaré - 1984 |

63 | The standard additive coalescent
- Aldous, Pitman
- 1998
(Show Context)
Citation Context ...rocess of ranked frequencies (f(t); ts0) in the improper cases\Gamma1 ! 1. Theorem 27 characterizes the entrance boundary of the ranked mass coalescent in both the proper and improper cases. See also =-=[1, 4, 5, 15]-=- regarding similar entrance laws and the entrance boundary for some particular binary coalescents with time parameter set (\Gamma1; 1) instead of (0; 1). 2.3 The two-parameter family The following two... |

62 |
The representation of partition structures
- Kingman
- 1978
(Show Context)
Citation Context ...ibuted according to some probability distribution on the line whose nth largest atom is x n , and whose continuous component has probability 1 \Gamma P n x n . 36 Theorem 36 (Kingman's correspondence =-=[24, 25]-=-). A bijective correspondence p $ P between probability distributions p of exchangeable random partitions of N and probability distributions P on S # is determined as follows. Each block B n of an exc... |

44 | Construction of Markovian coalescents
- Evans, Pitman
- 1998
(Show Context)
Citation Context ...] and Lushnikov [28]. See Aldous [3] for a recent survey of the scientific literature of these models and their relation to Smoluchowski's mean-field theory of coagulation phenomena. Evans and Pitman =-=[15]-=- gave a general framework for the rigorous construction of partition valued and discrete measure valued coalescent Markov processes allowing infinitely many massses, and treated the binary coalescent ... |

42 |
A mathematical reformulation of Derrida’s
- Ruelle
- 1987
(Show Context)
Citation Context ...escent in which every pair of blocks coalesces at rate 1. The cases= U , the uniform distribution on [0; 1] yields the coalescent derived by Bolthausen-Sznitman [9] from Ruelle's probability cascades =-=[39]-=-. See also [8] for another derivation of this coalescent from the genealogy of a continuous-state branching process. The rest of this paper is organized as follows. Section 2 describes the main result... |

38 | Partition structures derived from Brownian motion and stable subordinators
- Pitman
- 1995
(Show Context)
Citation Context ...m which shows how certain operations of coagulation and fragmentation act on the two-parameter family of distributions of exchangeable random partitions of N introduced in [31] and studied further in =-=[33, 34]-=-. Section 2.4 applies this theorem to the U-coalescent to recover some of the results of Bolthausen-Sznitman and to obtain various further developments. The conceptual framework of the paper is provid... |

37 | Tailfree and neutral random probabilities and their posterior distributions - Doksum - 1974 |

37 | Random discrete distributions invariant under size-biased permutation - Pitman - 1996 |

35 |
Exchangeability and the evolution of large populations
- Kingman
- 1982
(Show Context)
Citation Context ...bution. The PD(0; `) distribution has well known applications in population genetics, number theory, and combinatorics, as reviewed in [17, 6]. Formula (15) in this case is a variation due to Kingman =-=[26]-=- of the Ewens sampling formula [16], [20, Ch. 41]. See [30, 37] for interpretations of PD(ff; 0) in terms of excursions of a Markov process such as a Brownian motion or a recurrent Bessel process whos... |

34 |
The Bolthausen-Sznitman coalescent and the genealogy of continuous-state branching processes, Probability Theory and Related Fields 117
- Bertoin, Gall
(Show Context)
Citation Context ... every pair of blocks coalesces at rate 1. The cases= U , the uniform distribution on [0; 1] yields the coalescent derived by Bolthausen-Sznitman [9] from Ruelle's probability cascades [39]. See also =-=[8]-=- for another derivation of this coalescent from the genealogy of a continuous-state branching process. The rest of this paper is organized as follows. Section 2 describes the main results, with pointe... |

34 |
Random partitions in population genetics
- Kingman
(Show Context)
Citation Context ...nking the jumps of a subordinator ( t ; 0sts1), that is an increasing process with stationary independent increments, with E exp(\Gamma s ) = exp ` \Gammas Z 1 0 (1 \Gamma e \Gammax )(dx) ' fors0: (i)=-=[22, 23]-=- If (dx) = `x \Gamma1 e \Gammax dx for ` ? 0, corresponding tos1 with the gamma(`) distribution P( 1 2 dx) = \Gamma(`) \Gamma1 x `\Gamma1 e \Gammax dx, then V has PD(0; `) distribution. (ii)[30] If (d... |

31 | Une extension multidimensionnelle de la loi de l’arc sinus - Barlow, Pitman, et al. - 1989 |

20 | Random processes - Rosenblatt - 1974 |

19 |
Coagulation in finite systems
- Lushnikov
- 1978
(Show Context)
Citation Context ... by N.S.F. Grant DMS97-03961 1 1 Introduction Markovian coalescent models for the evolution of a system of masses by a random process of binary collisions were introduced by Marcus [29] and Lushnikov =-=[28]-=-. See Aldous [3] for a recent survey of the scientific literature of these models and their relation to Smoluchowski's mean-field theory of coagulation phenomena. Evans and Pitman [15] gave a general ... |

19 |
Stochastic coalescence
- Marcus
- 1968
(Show Context)
Citation Context ...h supported in part by N.S.F. Grant DMS97-03961 1 1 Introduction Markovian coalescent models for the evolution of a system of masses by a random process of binary collisions were introduced by Marcus =-=[29]-=- and Lushnikov [28]. See Aldous [3] for a recent survey of the scientific literature of these models and their relation to Smoluchowski's mean-field theory of coagulation phenomena. Evans and Pitman [... |

18 |
Sufficient statistics and extreme
- DYNKIN
- 1978
(Show Context)
Citation Context ...t have the same distribution for each t ? 0, and the conclusion follows. 2 Consider now the problem of characterizing all entrance laws (q t ; t ? 0) for the ranked mass coalescent. By general theory =-=[14]-=-, each entrance law is an integral mixture over some set of extreme entrance laws, called the entrance boundary of the semigroup. The following theorem identifies this entrance boundary with S # ifs\G... |

18 |
Population genetics theory - the past and the future
- Ewens
- 1990
(Show Context)
Citation Context ...th a stable distribution of index ff, then V has PD(ff; 0) distribution. The PD(0; `) distribution has well known applications in population genetics, number theory, and combinatorics, as reviewed in =-=[17, 6]-=-. Formula (15) in this case is a variation due to Kingman [26] of the Ewens sampling formula [16], [20, Ch. 41]. See [30, 37] for interpretations of PD(ff; 0) in terms of excursions of a Markov proces... |

14 | Random discrete distributions derived from self-similar random sets, Electron
- Pitman, Yor
- 1996
(Show Context)
Citation Context ...f the previous lemma. But the action of p -frag on S # is much simpler: this is just the operation described in a particular case in Corollary 13, and considered more generally as an action on S # in =-=[35]-=-. Lemma 35 Let \Pi i 1 for i = 1; 2 be two random partitions of N, with restrictions \Pi i n to [n]. Let p 2 andsp be two exchangeable probability distributions on P1 . Then the following two conditio... |

14 | On the relative lengths of excursions derived from a stable subordinator - Pitman, Yor - 1997 |

13 | The entrance boundary of the multiplicative coalescent
- Aldous, Limic
- 1998
(Show Context)
Citation Context ...rocess of ranked frequencies (f(t); ts0) in the improper cases\Gamma1 ! 1. Theorem 27 characterizes the entrance boundary of the ranked mass coalescent in both the proper and improper cases. See also =-=[1, 4, 5, 15]-=- regarding similar entrance laws and the entrance boundary for some particular binary coalescents with time parameter set (\Gamma1; 1) instead of (0; 1). 2.3 The two-parameter family The following two... |

13 |
Random combinatorial structures and prime factorizations
- Arratia, Barbour, et al.
- 1997
(Show Context)
Citation Context ...th a stable distribution of index ff, then V has PD(ff; 0) distribution. The PD(0; `) distribution has well known applications in population genetics, number theory, and combinatorics, as reviewed in =-=[17, 6]-=-. Formula (15) in this case is a variation due to Kingman [26] of the Ewens sampling formula [16], [20, Ch. 41]. See [30, 37] for interpretations of PD(ff; 0) in terms of excursions of a Markov proces... |

13 |
Coherent random allocations, and the Ewens-Pitman formula
- Kerov
- 2005
(Show Context)
Citation Context ...n i ) (65) 35 for some sequences of weights (b 1 ; b 2 ; : : :) and (w 1 ; w 2 ; : : :) and some sequence of normalization constantss(c 1 ; c 2 ; : : :) determined by these weights. As shown by Kerov =-=[21]-=-, the (ff; `) formula (15) and its limiting cases yield every EPF of this form. So it might be that Theorem 12 describes the only possible choices of non-degenerate laws p 1 and p of exchangeable rand... |

12 |
Non-self averaging effects in sum of random variables, in: On Three
- Derrida
- 1994
(Show Context)
Citation Context ...pretations of PD(ff; 0) in terms of excursions of a Markov process such as a Brownian motion or a recurrent Bessel process whose zero set is the closed range of a stable subordinator of index ff, and =-=[39, 10, 11, 9]-=- for applications of PD(ff; 0) in mathematical physics. Definition 11 For each probability measure p on P1 , define a Markov kernel p -frag on P1 , the p-fragmentation kernel as follows. Let p -frag(;... |

11 | Statistical properties of randomly broken objects and of multivalley structures in disordered systems - Derrida, Flyvbjerg - 1987 |

4 |
Distribution of the Mean Value for Certain Random Measures
- Tsilevich
- 1997
(Show Context)
Citation Context ...=2 (x) = 1 for all x. No explicit formula for g ff;p (x) seems to be known for other values of (ff; p), but one should be obtainable from (58) by inversion of the Mellin transform. See also Tsilevich =-=[41]-=- for study of related distributions. As a check on Corollary 33, formula (58) can be derived from (59) as follows. For k = 0; 1; 2; : : : a random variable Z r;s with beta(r; s) distribution has kth m... |

3 |
Coagulation in nite systems
- Lushnikov
- 1978
(Show Context)
Citation Context ... by N.S.F. Grant DMS97-03961 1s1 Introduction Markovian coalescent models for the evolution of a system of masses by a random process of binary collisions were introduced by Marcus [29] and Lushnikov =-=[28]-=-. See Aldous [3] for a recent survey of the scienti c literature of these models and their relation to Smoluchowski's mean- eld theory of coagulation phenomena. Evans and Pitman [15] gave a general fr... |

1 |
Non-self-averaging e ects in sums of random variables, spin glasses, random maps and random walks
- Derrida
- 1994
(Show Context)
Citation Context ...interpretations of PD( ; 0) in terms of excursions of a Markov process such asaBrownian motion or a recurrent Bessel process whose zero set is the closed range of a stable subordinator of index , and =-=[39, 10, 11, 9]-=- for applications of PD( ; 0) in mathematical physics. De nition 11 For each probability measure p on P1, de ne a Markov kernel p -frag on P1, the p-fragmentation kernel as follows. Let p -frag( ; ) b... |

1 |
cient statistics and extreme
- Su
- 1978
(Show Context)
Citation Context ...must have the same distribution for each t>0, and the conclusion follows. 2 Consider now the problem of characterizing all entrance laws (qt;t > 0) for the ranked mass - coalescent. By general theory =-=[14]-=-, each entrance law isanintegral mixture over some set of extreme entrance laws, called the entrance boundary of the semigroup. The following theorem identi es this entrance boundary with S # if ,1 < ... |