Consider a $Λ$-coalescent that comes down from infinity (meaning that it starts from a configuration containing infinitely many blocks at time 0, yet it has a finite number $N_t$ of blocks at any positive time $t>0$). We exhibit a deterministic function $v:(0,\infty)\to(0,\infty)$ such that $N_t/v(t)\to1$, almost surely, and in $L^p$ for any $p\geq1$, as $t\to0$. Our approach relies on a novel martingale technique.
We study topological properties of random metric spaces which arise by Lambda-coalescents. These are stochastic processes, which start with an infinite number of lines and evolve through multiple mergers in an exchangeable setting. We show that the resulting Lambda-coalescent measure tree is compact iff the Lambda-coalescent comes down from infinity, i.e. only consists of finitely many lines at any positive time. If the Lambda-coalescent stays infinite, the resulting metric measure space is not even locally compact. Our results are based on general notions of compact and locally compact (isome…
The $\X$-coalescent processes were initially studied by Möhle and Sagitov (2001), and introduced by Schweinsberg (2000) in their full generality. They arise in the mathematical population genetics as the complete class of scaling limits for genealogies of Cannings' models. The $\X$-coalescents generalize $Λ$-coalescents, where now simultaneous multiple collisions of blocks are possible. The standard version starts with infinitely many blocks at time 0, and it is said to come down from infinity if its number of blocks becomes immediately finite, almost surely. This work builds on the technique …
In this paper we look at the asymptotic number of r-caterpillars for $Λ$-coalescents which come down from infinity, under a regularly varying assumption. An r-caterpillar is a functional of the coalescent process started from $n$ individuals which, roughly speaking, is a block of the coalescent at some time, formed by one line of descend to which r-1 singletons have merged one by one. We show that the number of $r$-caterpillars, suitably scaled, converge to an explicit constant as the sample size n goes to infinity.
The nested Kingman coalescent describes the ancestral tree of a population undergoing neutral evolution at the level of individuals and at the level of species, simultaneously. We study the speed at which the number of lineages descends from infinity in this hierarchical coalescent process and prove the existence of an early-time phase during which the number of lineages at time $t$ decays as $ 2γ/ct^2$, where $c$ is the ratio of the coalescence rates at the individual and species levels, and the constant $γ\approx 3.45$ is derived from a recursive distributional equation for the number of lin…
This paper extends the notion of the $\la$-coalescent of Pitman (1999) to the spatial setting. The partition elements of the spatial $Λ$-coalescent migrate in a (finite) geographical space and may only coalesce if located at the same site of the space. We characterize the $Λ$-coalescents that come down from infinity, in an analogous way to Schweinsberg (2000). Surprisingly, all spatial coalescents that come down from infinity, also come down from infinity in a uniform way. This enables us to study space-time asymptotics of spatial $Λ$-coalescents on large tori in $d\ge 3$ dimensions. Our resul…
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…
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 infinity'. Moreover, of the many different (exchangeable) stochastic coalescent models, Kingman's coalescent is the `fastest' to come down from infinity. In this article we study what happens when we counteract this `fastest' coalescent with the action of an extreme form of fragmentation. We augment Kingman's coalescent, where any two blocks merge at rate $…
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…
We provide scaling limits for the block counting process and the fixation line of $Λ$-coalescents as the initial state $n$ tends to infinity under the assumption that the measure $Λ$ on $[0,1]$ satisfies $\int_{[0,1]}u^{-1}(Λ-bλ)({\rm d}u)<\infty$ for some $b>0$. Here $λ$ denotes the Lebesgue measure. The main result states that the block counting process, properly logarithmically scaled, converges in the Skorohod space to an Ornstein--Uhlenbeck type process as $n$ tends to infinity. The result is applied to beta coalescents with parameters $1$ and $b>0$. We split the generators into two parts…
We construct an extension of the Lambda-coalescent to a spatial continuum and analyse its behaviour. Like the Lambda-coalescent, the individuals in our model can be separated into (i) a dust component and (ii) large blocks of coalesced individuals. We identify a five phase system, where our phases are defined according to changes in the qualitative behaviour of the dust and large blocks. We completely classify the phase behaviour, including necessary and sufficient conditions for the model to come down from infinity. We believe that two of our phases are new to Lambda-coalescent theory and dir…
We study the block counting process and the fixation line of exchangeable coalescents. Formulas for the infinitesimal rates of both processes are provided. It is shown that the block counting process is Siegmund dual to the fixation line. For exchangeable coalescents restricted to a sample of size n and with dust we provide a convergence result for the block counting process as n tends to infinity. The associated limiting process is related to the frequencies of singletons of the coalescent. Via duality we obtain an analog convergence result for the fixation line of exchangeable coalescents wi…
The Ancestral Selection Graph (ASG) is an important genealogical process which extends the well-known Kingman coalescent to incorporate natural selection. We show that the number of lineages of the ASG with and without mutation is asymptotic to $2/t$ as $t\to 0$, in agreement with the limiting behaviour of the Kingman coalescent. We couple these processes on the same probability space using a Poisson random measure construction that allows us to precisely compare their hitting times. These comparisons enable us to characterise the speed of coming down from infinity of the ASG as well as its fl…
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.
A well-established model for the genealogy of a large population in equilibrium is Kingman's coalescent. For the population together with its genealogy evolving in time, this gives rise to a time-stationary tree-valued process. We study the sum of the branch lengths, briefly denoted as tree length, and prove that the (suitably compensated) sequence of tree length processes converges, as the population size tends to infinity, to a limit process with cadlag paths, infinite infinitesimal variance, and a Gumbel distribution as its equilibrium.
The block counting process and the fixation line of the Bolthausen-Sznitman coalescent are analyzed. Spectral decompositions for their generators and transition probabilities are provided leading to explicit expressions for functionals such as hitting probabilities and absorption times. It is furthermore shown that the block counting process and the fixation line of the Bolthausen-Sznitman $n$-coalescent, properly scaled, converge in the Skorohod topology to the Mittag-Leffler process and Neveu's continuous-state branching process respectively as the sample size $n$ tends to infinity. Strong r…
We consider the number of blocks involved in the last merger of a $Λ$-coalescent started with $n$ blocks. We give conditions under which, as $n \to \infty$, the sequence of these random variables a) is tight, b) converges in distribution to a finite random variable or c) converges to infinity in probability. Our conditions are optimal for $Λ$-coalescents that have a dust component. For general $Λ$, we relate the three cases to the existence, uniqueness and non-existence of quasi-invariant measures for the dynamics of the block-counting process, and in case b) investigate the time-reversal of t…
The block counting process with initial state $n$ counts the number of blocks of an exchangeable coalescent ($Ξ$-coalescent) restricted to a sample of size $n$. This work provides scaling limits for the block counting process of regular $Ξ$-coalescents that stay infinite, including $Ξ$-coalescents with dust and a large class of dust-free $Ξ$-coalescents. The main convergence result states that the block counting process, properly logarithmically scaled, converges in the Skorohod space to an Ornstein--Uhlenbeck type process as $n$ tends to infinity. The existence of such a scaling depends on a …
We consider standard $\La$-coalescents (or coalescents with multiple collisions) with a non-trivial "Kingman part". Equivalently, the driving measure $Λ$ has an atom at $0$; $Λ(\{0\})=c>0$. It is known that all such coalescents come down from infinity. Moreover, the number of blocks $N_t$ is asymptotic to $v(t) = 2/(ct)$ as $t\to 0$. In the present paper we investigate the second-order asymptotics of $N_t$ in the functional sense at small times. This complements our earlier results on the fluctuations of the number of blocks for a class of regular $\La$-coalescents without the Kingman part. In…
Consider a multitype coalescent process in which each block has a colour in $\{1,\ldots,d\}$. Individual blocks may change colour, and some number of blocks of various colours may merge to form a new block of some colour. We show that if the law of a multitype coalescent process is invariant under permutations of blocks of the same colour, has consistent Markovian projections, and has asychronous mergers, then it is a multitype $Λ$-coalescent: a process in which single blocks may change colour, two blocks of like colour may merge to form a single block of that colour, or large mergers across v…
We consider the genealogy of a sample of individuals taken from a spatially structured population when the variance of the offspring distribution is relatively large. The space is structured into discrete sites of a graph G. If the population size at each site is large, spatial coalescents with multiple mergers, so called spatial Lambda-coalescents, for which ancestral lines migrate in space and coalesce according to some Lambda-coalescent mechanism, are shown to be appropriate approximations to the genealogy of a sample of individuals. We then consider as the graph G the two dimensional torus…
We describe a representation of the Bolthausen-Sznitman coalescent in terms of the cutting of random recursive trees. Using this representation, we prove results concerning the final collision of the coalescent restricted to [n]: we show that the distribution of the number of blocks involved in the final collision converges as n tends to infinity, and obtain a scaling law for the sizes of these blocks. We also consider the discrete-time Markov chain giving the number of blocks after each collision of the coalescent restricted to [n]; we show that the transition probabilities of the time-revers…
The evolving Kingman coalescent is the tree-valued process which records the time evolution undergone by the genealogies of Moran populations. We consider the associated process of total external tree length of the evolving Kingman coalescent and its asymptotic behaviour when the number of leaves of the tree tends to infinity. We show that on the time-scale of the Moran model slowed down by a factor equal to the population size, the (centred and rescaled) external length process converges to a stationary Gaussian process with almost surely continuous paths and covariance function $c(s,t)=\Big(…
The Mittag-Leffler process $X=(X_t)_{t\ge 0}$ is introduced. This Markov process has the property that its marginal random variables $X_t$ are Mittag-Leffler distributed with parameter $e^{-t}$, $t\in [0,\infty)$, and the semigroup $(T_t)_{t\ge 0}$ of $X$ satisfies $T_tf(x)={\mathbb E}(f(x^{e^{-t}}X_t))$ for all $x\ge 0$ and all bounded measurable functions $f:[0,\infty)\to{\mathbb R}$. Further characteristics of the process $X$ are derived, for example an explicit formula for the joint moments of its finite dimensional distributions. The main result states that the block counting process of t…
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 …
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 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 the compact space of pairs of nested partitions of $\mathbb N$, where by analogy with models used in molecular evolution, we call "gene partition" the finer partition and "species partition" the coarser one. We introduce the class of nondecreasing processes valued in nested partitions, assumed Markovian and with exchangeable semigroup. These processes are said simple when each partition only undergoes one coalescence event at a time (but possibly the same time). Simple nested exchangeable coalescent (SNEC) processes can be seen as the extension of $Λ$-coalescents to nested partitio…
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…
Consider a system of Brownian particles on the real line where each pair of particles coalesces at a certain rate according to their intersection local time. Assume that there are infinitely many initial particles in the system. We give a necessary and sufficient condition for the number of particles to come down from infinity. We also identify the rate of this coming down from infinity for different initial configurations.
Consider a standard ${Λ}$-coalescent that comes down from infinity. Such a coalescent starts from a configuration consisting of infinitely many blocks at time $0$, but its number of blocks $N_t$ is a finite random variable at each positive time $t$. Berestycki et al. [Ann. Probab. 38 (2010) 207-233] found the first-order approximation $v$ for the process $N$ at small times. This is a deterministic function satisfying $N_t/v_t\to1$ as $t\to0$. The present paper reports on the first progress in the study of the second-order asymptotics for $N$ at small times. We show that, if the driving measure…
We consider a pure death process $(Z(t), t\ge0)$ with death rates $λ_n$ satisfying the condition $\sum_{n=2}^\infty λ_n^{-1}<\infty$ of coming from infinity, $Z(0)=\infty$, down to an absorbing state $n=1$. We establish limit theorems for $Z(t)$ as $t\to0$, which strengthen the results that can be extracted from [1]. We also prove a large deviation theorem assuming that $λ_n$ regularly vary as $n\to\infty$ with an index $ β>1$. It generalises a similar statement with $β=2$ obtained in [4] for $λ_n={n\choose 2}$.
We finely describe the speed of "coming down from infinity" for birth and death processes which eventually become extinct. Under general assumptions on the birth and death rates, we firstly determine the behavior of the successive hitting times of large integers. We put in light two different regimes depending on whether the mean time for the process to go from $n+1$ to $n$ is negligible or not compared to the mean time to reach $n$ from infinity. In the first regime, the coming down from infinity is very fast and the convergence is weak. In the second regime, the coming down from infinity is …
We consider the critical Galton-Watson process with overlapping generations stemming from a single founder. Assuming that both the variance of the offspring number and the average generation length are finite, we establish the convergence of the finite-dimensional distributions, conditioned on non-extinction at a remote time of observation. The limiting process is identified as a pure death process coming down from infinity. This result brings a new perspective on Vatutin's dichotomy claiming that in the critical regime of age-dependent reproduction, an extant population either contains a larg…
We approximate stochastic processes in finite dimension by dynamical systems. We provide trajectorial estimates which are uniform with respect to the initial condition for a well chosen distance. This relies on some non-expansivity property of the flow, which allows to deal with non-Lipschitz vector fields. We use the stochastic calculus and follow the martingale technics initiated in Berestycki and al [5] to control the fluctuations. Our main applications deal with the short time behavior of stochastic processes starting from large initial values. We state general properties on the coming dow…
In this note, we are interested on the event of extinction and the property of coming down from infinity of continuous state branching (or CB for short) processes with competition in a Lévy environment whose branching mechanism satisfies the so-called Grey's condition. In particular, we deduce, under the assumption that the Lévy environment does not drift towards infinity, that for any starting point the process becomes extinct in finite time a.s. Moreover if we impose an integrability condition on the competition mechanism, then the process comes down from infinity regardless the long term be…
We consider the class of exchangeable fragmentation-coagulation (EFC) processes where coagulations are multiple and not simultaneous, as in a $Λ$-coalescent, and fragmentation dislocates at finite rate an individual block into sub-blocks of infinite size. We call these partition-valued processes, simple EFC processes, and study the question whether such a process, when started with infinitely many blocks, can visit partitions with a finite number of blocks or not. When this occurs, one says that the process comes down from infinity. We introduce two sharp parameters $θ_{\star}\leq θ^{\star}\in…
We identify a new natural coalescent structure, which we call the seed-bank coalescent, that describes the gene genealogy of populations under the influence of a strong seed-bank effect, where "dormant forms" of individuals (such as seeds or spores) may jump a significant number of generations before joining the "active" population. Mathematically, our seed-bank coalescent appears as scaling limit in a Wright-Fisher model with geometric seed-bank age structure if the average time of seed dormancy scales with the order of the total population size $N$. This extends earlier results of Kaj, Krone…
We finely describe the "coming down from infinity" for birth and death processes which eventually become extinct. Our biological motivation is to study the decrease of regulated populations which are initially large. Under general assumptions on the birth and death rates, we describe the behavior of the hitting time of large integers. We let two regimes appear and derive an expression of the speed of coming down from infinity. In the case of death rates with regular variations, we also get a central limit theorem and the asymptotic probability of extinction in small times. Finally, we apply ou…
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…
Using Foster-Lyapunov techniques we establish new conditions on non-extinction, non-explosion, coming down from infinity and staying infinite, respectively, for the general continuous-state nonlinear branching processes introduced in Li et al. (2019). These results can be applied to identify boundary behaviors for the critical cases of the above nonlinear branching processes with power rate functions driven by Brownian motion and (or) stable Poisson random measure, which was left open in Li et al. (2019). In particular, we show that even in the critical cases, a phase transition happens betwee…
In this paper we consider diffusions on the half line (0, $\infty$) such that the expectation of the arrival time at the origin is uniformly bounded in the initial point. This implies that there is a well defined diffusion process starting from infinity, which takes finite values at positive times. We study the behaviour of hitting times of large barriers and in a dual way, the behaviour of the process starting at infinity for small time. In particular we prove that the process coming down from infinity is in small time governed by a specific deterministic function. Suitably normalized fluctua…
Consider a spectrally positive Lévy process $Z$ with log-Laplace exponent $Ψ$ and a positive continuous function $R$ on $(0,\infty)$. We investigate the entrance from $\infty$ of the process $X$ obtained by changing time in $Z$ with the inverse of the additive functional $η(t)=\int_{0}^{t}\frac{{\rm d} s}{R(Z_s)}$. We provide a necessary and sufficient condition for infinity to be an entrance boundary of the process $X$. Under this condition, the process can start from infinity and we study its speed of coming down from infinity. When the Lévy process has a negative drift $δ:=-γ<0$, sufficient…
We consider the behavior of the Gaussian concentration bound (GCB) under stochastic time evolution. More precisely, we consider a Markovian diffusion process on $\mathbb{R}^d$ and start the process from an initial distribution $μ$ that satisfies GCB. We then study the question whether GCB is preserved under the time-evolution, and if yes, how the constant behaves as a function of time. In particular, if for the constant we obtain a uniform bound, then we can also conclude properties of the stationary measure(s) of the diffusion process. This question, as well as the methodology developed in th…
