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 $\int_0^1x^{k-2}(1-x)^{b-k}\varLambda(\mathrm{d}x)$. It has recently been shown that if $1<α<2$, the $\varLambda$-coalescent in which $\varLambda$ is the $\operatorname {Beta}(2-α,α)$ distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an $α$-stable branching mechanism. Here we use facts about CSBPs to establish new results about the small-time asymptotics of beta coalescents. We prove an a.s. limit theorem for the number of blocks at small times, and we establish results about the sizes of the blocks. We also calculate the Hausdorff and packing dimensions of a metric space associated with the beta coalescents, and we find the sum of the lengths of the branches in the coalescent tree, both of which are determined by the behavior of coalescents at small times. We extend most of these results to other $\varLambda$-coalescents for which $\varLambda$ has the same asymptotic behavior near zero as the $\operatorname {Beta}(2-α,α)$ distribution. This work complements recent work of Bertoin and Le Gall, who also used CSBPs to study small-time properties of $\varLambda$-coalescents.
We show that the total number of collisions in the exchangeable coalescent process driven by the beta $(1,b)$ measure converges in distribution to a 1-stable law, as the initial number of particles goes to infinity. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instance $b=1$, which corresponds to the Bolthausen--Sznitman coalescent. The approach we take is based on estimating the quality of a renewal approximation to the coalescent in terms of a suitable Wasserstein distance. Application of the method to beta…
The Beta coalescents are stochastic processes modeling the genealogy of a population. They appear as the rescaled limits of the genealogical trees of numerous stochastic population models. In this article, we take interest in the number of blocs at small times in the Beta coalescent. Berestycki, Berestycki and Schweinsberg proved a law of large numbers for this quantity. Recently, Limic and Talarczyk proved that a functional central limit theorem holds as well. We give here a simple proof for an unidimensional version of this result, using a coupling between Beta coalescents and continuous-tim…
In mathematical population genetics, it is well known that one can represent the genealogy of a population by a tree, which indicates how the ancestral lines of individuals in the population coalesce as they are traced back in time. As the population evolves over time, the tree that represents the genealogy of the population also changes, leading to a tree-valued stochastic process known as the evolving coalescent. Here we will consider the evolving coalescent for populations whose genealogy can be described by a beta coalescent, which is known to give the genealogy of populations with very la…
We consider Beta$(2-α, α)$-coalescents with parameter range $1 <α<2$ starting from $n$ leaves. The length $\ell^{(n)}_r$ of order $r$ in the $n$-Beta$(2-α, α)$-coalescent tree is defined as the sum of the lengths of all branches that carry a subtree with $r$ leaves. We show that for any $s \in \mathbb N$ the vector of suitably centered and rescaled lengths of orders $1\le r \le s$ converges in distribution to a multivariate stable distribution as the number of leaves tends to infinity.
We describe a simple construction of Kingman's coalescent in terms of a Brownian excursion. This construction is closely related to, and sheds some new light on, earlier work by Aldous and Warren. Our approach also yields some new results: for instance, we obtain the full multifractal spectrum of Kingman's coalescent. This complements earlier work on Beta-coalescents by the authors and Schweinsberg. Surprisingly, the thick part of the spectrum is not obtained by taking the limit as $α\to 2$ in the result for Beta-coalescents mentioned above. Other analogies and differences between the case of …
For $1<α<2$ we derive the asymptotic distribution of the total length of {\em external} branches of a Beta$(2-α, α)$-coalescent as the number $n$ of leaves becomes large. It turns out the fluctuations of the external branch length follow those of $τ_n^{2-α}$ over the entire parameter regime, where $τ_n$ denotes the random number of coalescences that bring the $n$ lineages down to one. This is in contrast to the fluctuation behavior of the total branch length, which exhibits a transition at $α_0 = (1+\sqrt 5)/2$.
This article shows the asymptotics of distributions of various functionals of the Beta$(2-α,α)$ $n$-coalescent process with $1<α<2$ when $n$ goes to infinity. This process is a Markov process taking {values} in the set of partitions of $\{1, \dots, n\}$, evolving from the intial value $\{1\},\cdots, \{n\}$ by merging (coalescing) blocks together into one and finally reaching the absorbing state $\{1, \dots, n\}$. The minimal clade of $1$ is the block which contains $1$ at the time of coalescence of the singleton $\{1\}$. The limit size of the minimal clade of $1$ is provided. To this, we expre…
We quantify the manner in which the beta coalescent $Π=\{ Π(t), t\geq 0\},$ with parameters $a\in (0, 1),$ $b>0,$ comes down from infinity. Approximating $Π$ by its restriction $Π^n$ to $[n]\:= \{1, \ldots, n\},$ the suitably rescaled block counting process $n^{-1}\#Π^n(tn^{a-1})$ has a deterministic limit, $c(t)$, as $n\to\infty.$ An explicit formula for $c(t)$ is provided in Theorem 1. The block size spectrum $(\mathfrak{c}_{1}Π^n(t), \ldots, \mathfrak{c}_{n}Π^n(t)),$ where $\mathfrak{c}_{i}Π^n(t)$ counts the number of blocks of size $i$ in $Π^n(t),$ captures more refined information about t…
In this paper, we consider Beta$(2-α,α)$ (with $1<α<2$) and related $Λ$-coalescents. If $T^{(n)}$ denotes the length of an external branch of the $n$-coalescent, we prove the convergence of $n^{α-1}T^{(n)}$ when $n$ tends to $ \infty $, and give the limit. To this aim, we give asymptotics for the number $σ^{(n)}$ of collisions which occur in the $n$-coalescent until the end of the chosen external branch, and for the block counting process associated with the $n$-coalescent.
We derive the asymptotic distribution of the total length $L_n$ of a $\operatorname {Beta}(2-α,α)$-coalescent tree for $1<α<2$, starting from $n$ individuals. There are two regimes: If $α\le1/2(1+\sqrt{5})$, then $L_n$ suitably rescaled has a stable limit distribution of index $α$. Otherwise $L_n$ just has to be shifted by a constant (depending on $n$) to get convergence to a nondegenerate limit distribution. As a consequence, we obtain the limit distribution of the number $S_n$ of segregation sites. These are points (mutations), which are placed on the tree's branches according to a Poisson p…
We apply recently developed inference methods based on general coalescent processes to DNA sequence data obtained from various marine species. Several of these species are believed to exhibit so-called shallow gene genealogies, potentially due to extreme reproductive behaviour, e.g. via Hedgecock's "reproduction sweepstakes". Besides the data analysis, in particular the inference of mutation rates and the estimation of the (real) time to the most recent common ancestor, we briefly address the question whether the genealogies might be adequately described by so-called Beta coalescents (as oppos…
In this paper we obtain scaling limits of $Λ$-coalescents near time zero under a regularly varying assumption. In particular this covers the case of Kingman's coalescent and beta coalescents. The limiting processes are coalescents with infinite mass, obtained geometrically as tangent cones of Evans metric space associated with the coalescent. In the case of Kingman's coalescent we are able to obtain a simple construction of the limiting space using a two-sided Brownian motion.
We consider a $Λ$-coalescent and we study the asymptotic behavior of the total length $L^{(n)}_{ext}$ of the external branches of the associated $n$-coalescent. For Kingman coalescent, i.e. $Λ=δ_0$, the result is well known and is useful, together with the total length $L^{(n)}$, for Fu and Li's test of neutrality of mutations% under the infinite sites model asumption . For a large family of measures $Λ$, including Beta$(2-α,α)$ with $0<α<1$, M{ö}hle has proved asymptotics of $L^{(n)}_{ext}$. Here we consider the case when the measure $Λ$ is Beta$(2-α,α)$, with $1<α<2$. We prove that $n^{α-2}L…
Coalescents with multiple collisions, also known as $Λ$-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 blocks can merge at the same time to form a single block. In the case that the measure $Λ$ is the $\operatorname {Beta}(2-α,α)$ distribution, they are also known to describe the genealogies of large populations where a single individual can produce a large number of offspring. Here, we use a recent result of Birkner et al. to prove that Beta-coalescents can be embedd…
We study a class of coalescents derived from a sampling procedure out of N i.i.d. Pareto(alpha) random variables, normalized by their sum, including beta-size-biasing on total length effects (beta < alpha). Depending on the range of alpha, we derive the large N limit coalescents structure, leading either to a discrete-time Poisson-Dirichlet(alpha,-beta) Xi-coalescent (alpha in [0,1)), or to a family of continuous-time Beta(2-alpha,alpha-beta) Lambda-coalescents (alpha in [1,2)), or to the Kingman coalescent (alpha geq 2). We indicate that this class of coalescent processes (and their scaling l…
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 same vertex merge randomly according to a given coalescing mechanism. A remarkable property of mean-field coalescent processes is that they may come down from infinity, meaning that, starting with an infinite number of particles, only a finite number remains after any positive amount of time, almost surely. We show here however that, in the spatial setting, …
We consider diploid bi-parental analogues of Cannings models: in a population of fixed size $N$ the next generation is composed of $V_{i,j}$ offspring from parents $i$ and $j$, where $V=(V_{i,j})_{1\le i\neq j \le N}$ is a (jointly) exchangeable (symmetric) array. Every individual carries two chromosome copies, each of which is inherited from one of its parents. We obtain general conditions, formulated in terms of the vector of the total number of offspring to each individual, for the convergence of the properly scaled ancestral process for an $n$-sample of genes towards a ($Ξ$-)coalescent. Th…
Consider a population of fixed size that evolves over time. At each time, the genealogical structure of the population can be described by a coalescent tree whose branches are traced back to the most recent common ancestor of the population. As time goes forward, the genealogy of the population evolves, leading to what is known as an evolving coalescent. We will study the evolving coalescent for populations whose genealogy can be described by the Bolthausen-Sznitman coalescent. We obtain the limiting behavior of the evolution of the time back to the most recent common ancestor and the total le…
We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results to population dynamics where the genealogy is given by a Lambda-coalescent. This allows us to derive an exact formula for the asymptotic behavior of the site and allele frequency spectrum and the number of segregating sites, as the sample size tends to infinity. Some of our results hold in the case of a general Lambda-coalescent that comes down from infinity, but we obtain more precise information u…
When an advantageous mutation occurs in a population, the favorable allele may spread to the entire population in a short time, an event known as a selective sweep. As a result, when we sample $n$ individuals from a population and trace their ancestral lines backwards in time, many lineages may coalesce almost instantaneously at the time of a selective sweep. We show that as the population size goes to infinity, this process converges to a coalescent process called a coalescent with multiple collisions. A better approximation for finite populations can be obtained using a coalescent with simul…
This paper studies the spatial coalescent on $\Z^2$. In our setting, the partition elements are located at the sites of $\Z^2$ and undergo local delayed coalescence and migration. That is, pairs of partition elements located at the same site coalesce into one partition element after exponential waiting times. In addition, the partition elements perform independent random walks. The system starts in either locally finite configurations or in configurations containing countably many partition elements per site. These two situations are relevant if the coalescent is used to study the scaling limi…
We propose a new method of analyzing the asymptotics of moments of certain linear random recurrences which is based on the technique of iterative functions. By using the method, we show that the moments of the number of collisions and the absorption time in the Poisson-Dirichlet coalescent behave like the powers of the "log star" function which grows slower than any iteration of the logarithm, and thereby prove a weak law of large numbers. Finally, we discuss merits and limitations of the method and give several examples related to beta coalescents, recursive algorithms and random trees.
We describe a new general connection between $Λ$-coalescents and genealogies of continuous-state branching processes. This connection is based on the construction of an explicit coupling using a particle representation inspired by the lookdown process of Donnelly and Kurtz. This coupling has the property that the coalescent comes down from infinity if and only if the branching process becomes extinct, thereby answering a question of Bertoin and Le Gall. The coupling also offers new perspective on the speed of coming down from infinity and allows us to relate power-law behavior for $N^Λ(t)$ to …
Consider a continuous-state branching population constructed as a flow of nested subordinators. Inverting the subordinators and reversing time give rise to a flow of coalescing Markov processes (with negative jumps) which correspond to the ancestral lineages of individuals in the current generation. The process of the ancestral lineage of a fixed individual is the Siegmund dual process of the continuous-state branching process. We study its semi-group, its long-term behavior and its generator. In order to follow the coalescences in the ancestral lineages and to describe the backward genealogy …
We study the genealogy of a solvable population model with $N$ particles on the real line which evolves according to a discrete-time branching process with selection. At each time step, every particle gives birth to children around $a$ times its current position, where $a>0$ is a parameter of the model. Then, the $N$ rightmost new-born children are selected to form the next generation. We show that the genealogical trees of the process converge to those of a Beta coalescent as $N \to \infty$. The process we consider can be seen as a toy-model version of a continuous-time branching process with…
We derive the asymptotic behavior of the total, active and inactive branch lengths of the seed bank coalescent, when the size of the initial sample grows to infinity. Those random variables have important applications for populations evolving under some seed bank effects, such as plants and bacteria, and for some cases of structured populations like metapopulations. The proof relies on the study of the tree at a stopping time corresponding to the first time that a deactivated lineage reactivates. We also give conditional sampling formulas for the random partition and we study the system at the…
Motivated by stability questions on piecewise deterministic Markov models of bacterial chemotaxis, we study the long time behavior of a variant of the classic telegraph process having a non-constant jump rate that induces a drift towards the origin. We compute its invariant law and show exponential ergodicity, obtaining a quantitative control of the total variation distance to equilibrium at each instant of time. These results rely on an exact description of the excursions of the process away from the origin and on the explicit construction of an original coalescent coupling for both velocity …
The paper describes the global limiting behavior of Gaussian beta ensembles where the parameter $β$ is allowed to vary with the matrix size $n$. In particular, we show that as $n \to \infty$ with $nβ\to \infty$, the empirical distribution converges weakly to the semicircle distribution, almost surely. The Gaussian fluctuation around the limit is then derived by a martingale approach.
We study the limiting behavior of Gaussian beta ensembles in the regime where $βn = const$ as $n \to \infty$. The results are (1) Gaussian fluctuations for linear statistics of the eigenvalues, and (2) Poisson convergence of the bulk statistics. (2) is an alternative proof of the result by F.~Benaych-Georges and S.~Péché (2015) with the explicit form of the intensity measure.
There is a unique unitarily-invariant ensemble of $N\times N$ Hermitian matrices with a fixed set of real eigenvalues $a_1 > \dots > a_N$. The joint eigenvalue distribution of the $(N - 1)$ top-left principal submatrices of a random matrix from this ensemble is called the orbital unitary process. There are analogous matrix ensembles of symmetric and quaternionic Hermitian matrices that lead to the orbital orthogonal and symplectic processes, respectively. By extrapolation, on the dimension of the base field, of the explicit density formulas, we define the orbital beta processes. We prove the u…
Let $X_1,X_2,...$ be a sequence of random variables satisfying the distributional recursion $X_1=0$ and $X_n= X_{n-I_n}+1$ for $n=2,3,...$, where $I_n$ is a random variable with values in $\{1,...,n-1\}$ which is independent of $X_2,...,X_{n-1}$. The random variable $X_n$ can be interpreted as the absorption time of a suitable death Markov chain with state space ${\mathbb N}:=\{1,2,...\}$ and absorbing state 1, conditioned that the chain starts in the initial state $n$. This paper focuses on the asymptotics of $X_n$ as $n$ tends to infinity under the particular but important assumption that th…
Time series data may exhibit clustering over time and, in a multiple time series context, the clustering behavior may differ across the series. This paper is motivated by the Bayesian non--parametric modeling of the dependence between the clustering structures and the distributions of different time series. We follow a Dirichlet process mixture approach and introduce a new class of multivariate dependent Dirichlet processes (DDP). The proposed DDP are represented in terms of vector of stick-breaking processes with dependent weights. The weights are beta random vectors that determine different …
