Related papers: Multitype $\Lambda$-coalescents
We provide new connections between multitype $\Lambda$-coalescents and multitype continuous state branching processes via duality and a homeomorphism on their parameter space. The approach is based on a sequential sampling procedure for the…
$\Lambda$-coalescents model the evolution of a coalescing system in which any number of blocks randomly sampled from the whole may merge into a larger block. For the coalescent restricted to initially $n$ singletons we study the collision…
Coalescents with multiple collisions, also known as $\Lambda$-coalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several…
We consider a stochastic model, called the replicator coalescent, describing a system of blocks of $k$ different types which undergo pairwise mergers at rates depending on the block types: with rate $C_{i,j}$ blocks of type $i$ and $j$…
Birkner et al. obtained necessary and sufficient conditions for the frequency between two independent and identically distributed continuous-state branching processes time-changed by a functional of the total mass process to be a Markov…
Coalescents with multiple collisions (also called Lambda-coalescents or simple exchangeable coalescents) are used as models of genealogies. We study a new class of Markovian coalescent processes connected to a population model with…
We introduce a Poissonization method to study the coalescent structure of uniform samples from branching processes. This method relies on the simple observation that a uniform sample of size $k$ taken from a random set with positive…
We define a Markov process on the partitions of $[n]=\{1,\ldots,n\}$ by drawing a sample in $[n]$ at each time of a Poisson process, by merging blocks that contain one of these points and by leaving all other blocks unchanged. This…
We examine the total number of collisions $C_n$ in the $\Lambda$-coalescent process which starts with $n$ particles. A linear growth and a stable limit law for $C_n$ are shown under the assumption of a power-like behaviour of the measure…
For a finite measure $\varLambda$ on $[0,1]$, the $\varLambda$-coalescent is a coalescent process such that, whenever there are $b$ clusters, each $k$-tuple of clusters merges into one at rate…
We study several fundamental properties of a class of stochastic processes called spatial Lambda-coalescents. In these models, a number of particles perform independent random walks on some underlying graph G. In addition, particles on the…
Multiple-merger coalescents, e.g. $\Lambda$-$n$-coalescents, have been proposed as models of the genealogy of $n$ sampled individuals for a range of populations whose genealogical structures are not captured well by Kingman's…
We define a multi-type coalescent point process of a general branching process with finitely many types. This multi-type coalescent fully describes the genealogy of the (quasi-stationary) standing population, providing types along ancestral…
Bertoin and Le Gall (2003) introduced a certain probability measure valued Markov process that describes the evolution of a population, such that a sample from this population would exhibit a genealogy given by the so-called…
We consider the lambda-coalescent processes with positive frequency of singleton clusters. The class in focus covers, for instance, the beta$(a,b)$-coalescents with $a>1$. We show that some large-sample properties of these processes can be…
Consider the Markov process taking values in the partitions of N such that each pair of blocks merges at rate one, and each integer is eroded, i.e., becomes a singleton block, at rate d. This is a special case of exchangeable…
We study infinite systems of particles which undergo coalescence and fragmentation, in a manner determined solely by their masses. A pair of particles having masses $x$ and $y$ coalesces at a given rate $K(x,y)$. A particle of mass $x$…
We prove several limit theorems that relate coalescent processes to continuous-state branching processes. Some of these theorems are stated in terms of the so-called generalized Fleming-Viot processes, which describe the evolution of a…
We introduce the multiplicative coalescent with linear deletion, a continuous-time Markov process describing the evolution of a collection of blocks. Any two blocks of sizes $x$ and $y$ merge at rate $xy$, and any block of size $x$ is…
An important property of Kingman's coalescent is that, starting from a state with an infinite number of blocks, over any positive time horizon, it transitions into an almost surely finite number of blocks. This is known as `coming down from…