Results 1  10
of
20
Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
Abstract

Cited by 46 (3 self)
 Add to MetaCart
Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
Gibbs Fragmentation Trees
, 2008
"... We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbstype fragmentation tree with Aldous’ betasplitting model, which has an extended parameter range β>−2 with respect to the beta(β + 1,β + 1) probability distributions on which it is based. In the mul ..."
Abstract

Cited by 19 (6 self)
 Add to MetaCart
We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbstype fragmentation tree with Aldous’ betasplitting model, which has an extended parameter range β>−2 with respect to the beta(β + 1,β + 1) probability distributions on which it is based. In the multifurcating case, we show that Gibbs fragmentation trees are associated with the twoparameter Poisson–Dirichlet models for exchangeable random partitions of N, with an extended parameter range 0 ≤ α ≤ 1, θ ≥−2α and α<0, θ =−mα, m ∈ N.
Meinardus ’ theorem on weighted partitions: extensions and
, 2008
"... a probabilistic proof ..."
(Show Context)
Limit shapes of Gibbs distributions on the set of integer partitions: The expansive case,
 Ann. Inst. H. Poincar Probab. Statist.
, 2008
"... ..."
(Show Context)
Poisson representation of a Ewens fragmentation process
"... A simple explicit construction is provided of a partitionvalued fragmentation process whose distribution on partitions of [n] = {1,...,n} at time θ ≥ 0 is governed by the Ewens sampling formula with parameter θ. These partitionvalued processes are exchangeable and consistent, as n varies. They ca ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
(Show Context)
A simple explicit construction is provided of a partitionvalued fragmentation process whose distribution on partitions of [n] = {1,...,n} at time θ ≥ 0 is governed by the Ewens sampling formula with parameter θ. These partitionvalued processes are exchangeable and consistent, as n varies. They can be derived by uniform sampling from a corresponding mass fragmentation process defined by cutting a unit interval at the points of a Poisson process with intensity θx −1 dx on R+, arranged to be intensifying as θ increases. 1
Record indices and ageordered frequencies in Exchangeable Gibbs Partitions
, 2008
"... Abstract We consider a random partition Π of N = {1, 2,...} such that, for each n, its restriction Πn to [n] = {1,..., n} is given by an exchangeable Gibbs partition with parameters α, V for α ∈ (−∞, 1] and V = (Vn,k) defined recursively by setting V1,1 = 1 and Vn,k = (n − αk)Vn+1,k + Vn+1,k+1 k ≤ ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
(Show Context)
Abstract We consider a random partition Π of N = {1, 2,...} such that, for each n, its restriction Πn to [n] = {1,..., n} is given by an exchangeable Gibbs partition with parameters α, V for α ∈ (−∞, 1] and V = (Vn,k) defined recursively by setting V1,1 = 1 and Vn,k = (n − αk)Vn+1,k + Vn+1,k+1 k ≤ n = 1, 2,... (Gnedin and Pitman 2006). By ranking the blocks Πn1,..., Πnk of Πn by their ageorder i.e. by the order of their least elements i1,...,ik, we study how the distribution of the frequencies of the blocks depends on i1,...,ik. Several interesting representations for the limit ageordered relative frequencies X1, X2,... of Π arise, depending on which ij’s one conditions on. In particular, conditioning on the entire vector i = 1 = i1 < i2 <..., a representation is Xj = ξj−1 (1 − ξi) j = 1, 2,... i=j where the ξj’s are independent Beta random variables with parameters, respectively, (1−α, ij+1−αj−1). We show the connection of such a representation with the socalled BetaStacy class of random discrete distributions (Walker and Muliere 1997). The vector i is found to form a Markov chain depending on both α and V. When V is chosen from Pitman’s subfamily, the twoparameter GEM distribution is reobtained by averaging the ξ over i. Conditioning on ik alone, we give two alternative representations for the Laplace transform of both − log Xk and − log ( ∑k i=1 Xi), and we characterize Ewens ’ partitions as the only exchangeable Gibbs partitions for which − logXkik can be represented as an infinite sum of independent random variables. We finally show that, for every k, conditional on ∑k i=1 Xi, the distribution of the normalized ageordered frequencies X1 / ∑k i=1 Xi,..., Xk / ∑k i=1 Xi is a mixture of Dirichlet distributions on the (k − 1)dimensional simplex, whose mixing measure is indexed by ik. We provide a nontrivial explicit formula for the marginal distribution of ik. Many of the mentioned representations are extensions of Griffiths and Lessard (2005) results on Ewens ’ partitions.
Asymptotics of counts of small components in random structures and models of coagulation –fragmentation.
, 2008
"... ..."
On time dynamics of coagulationfragmentation processes
, 2008
"... 1 transient 2 We establish a characterization of coagulationfragmentation processes, such that the induced birth and death processes depicting the total number of groups at time t ≥ 0 are time homogeneous. Based on this, we provide a characterization of meanfield Gibbs coagulationfragmentation mod ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
1 transient 2 We establish a characterization of coagulationfragmentation processes, such that the induced birth and death processes depicting the total number of groups at time t ≥ 0 are time homogeneous. Based on this, we provide a characterization of meanfield Gibbs coagulationfragmentation models, which extends the one derived by Hendriks et al. As a by product of our results, the class of solvable models is widened and a question posed by N. Berestycki and Pitman is answered, under restriction to meanfield models. transient 3 1 Introduction, objective and the context The time dynamics of a time homogeneous Markov process X(t), t ≥ 0 on a space Ω = {η} of states η is described by the set of transition probabilities p ˜ ζ (η; t): = P(X(t) = η X(0) = ˜ ζ), ˜ ζ, η ∈ Ω, t ≥ 0.