%0 Journal Article %1 pitman1999multiple %A Pitman, Jim %D 1999 %J Ann. Probab. %K coalescent_theory combinatorics lambda_coalescent %N 4 %P 1870--1902 %T Coalescents with multiple collisions %U http://projecteuclid.org/euclid.aop/1022874819 %V 27 %X For each finite measure $Łambda$ on $0,1$, the $Łambda$-coalescent Markov process, with state space the compact set of all partitions of $N$, is constructed in such a way that the restriction of the partition to each finite subset of $N$ is a Markov chain, with the following transition rates: when the partition has $b$ blocks, each $k$-tuple of blocks is merging to form a single block at rate $ınt^1_0x^k-2(1-x)^b-kŁambda(dx)$ see also S. N. Evans and J. W. Pitman, Ann. Inst. H. Poincaré Probab. Statist. 34 (1998), no. 3, 339--383; MR1625867 (99k:60184); and S. Sagitov, J. Appl. Probab. 36 (1999), no. 4, 1116--1125; MR1742154 (2001f:92019). Examples include Kingman's coalescent, where $Łambda$ is the unit mass at $0$, and the Bolthausen-Sznitman coalescent, where $Łambda=U$, the uniform distribution on $0,1$. The two-parameter Poisson-Dirichlet family of random discrete distributions derived from a stable subordinator, and corresponding exchangeable random partitions of $N$ governed by a generalization of the Ewens sampling formula are applied to describe transition mechanisms for processes of coalescence and fragmentation, including the $U$-coalescent and its time reversal.

@article{pitman1999multiple, abstract = {For each finite measure $\Lambda$ on $[0,1]$, the $\Lambda$-coalescent Markov process, with state space the compact set of all partitions of $\bold N$, is constructed in such a way that the restriction of the partition to each finite subset of $\bold N$ is a Markov chain, with the following transition rates: when the partition has $b$ blocks, each $k$-tuple of blocks is merging to form a single block at rate $\int^1_0x^{k-2}(1-x)^{b-k}\Lambda(dx)$ [see also S. N. Evans and J. W. Pitman, Ann. Inst. H. Poincaré Probab. Statist. 34 (1998), no. 3, 339--383; MR1625867 (99k:60184); and S. Sagitov, J. Appl. Probab. 36 (1999), no. 4, 1116--1125; MR1742154 (2001f:92019)]. Examples include Kingman's coalescent, where $\Lambda$ is the unit mass at $0$, and the Bolthausen-Sznitman coalescent, where $\Lambda=U$, the uniform distribution on $[0,1]$. The two-parameter Poisson-Dirichlet family of random discrete distributions derived from a stable subordinator, and corresponding exchangeable random partitions of $\bold N$ governed by a generalization of the Ewens sampling formula are applied to describe transition mechanisms for processes of coalescence and fragmentation, including the $U$-coalescent and its time reversal.}, added-at = {2009-01-21T00:22:56.000+0100}, author = {Pitman, Jim}, biburl = {https://www.bibsonomy.org/bibtex/2b0c187d828aca9180a5d01313ef30a19/peter.ralph}, coden = {APBYAE}, fjournal = {The Annals of Probability}, interhash = {0f9616981e6dffe679e2f8b65ce9ea88}, intrahash = {b0c187d828aca9180a5d01313ef30a19}, issn = {0091-1798}, journal = {Ann. Probab.}, keywords = {coalescent_theory combinatorics lambda_coalescent}, mrclass = {60C05 (05A18 60G09 60G57)}, mrnumber = {MR1742892 (2001h:60016)}, mrreviewer = {Andrew D. Barbour}, number = 4, pages = {1870--1902}, timestamp = {2011-11-01T07:01:18.000+0100}, title = {Coalescents with multiple collisions}, url = {http://projecteuclid.org/euclid.aop/1022874819}, volume = 27, year = 1999 }