Second order behavior of the block counting process of beta coalescents

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-time branching processes.


Introduction
A coalescent process is a stochastic model for the genealogy of an infinite haploid population, built backward in time.In such a model, an individual is represented by an integer n ∈ N. At each time t, we denote by Π(t) the partition of N such that two individuals i and j belong to the same set in Π(t) (that we call "bloc" from now on) if they share a common ancestor less than t units of time in the past.In particular, we always assume that Π(0) = {{1}, {2}, . ..} is the partition in singletons.We construct (Π(t), t ≥ 0) as a Markov process on the set of partitions, that gets coarser over time.
Let Λ be a probability measure on [0, 1].The Λ-coalescent is a coalescent process such that given there are b distinct blocs in Π(t), any particular set of k blocs merge at rate The Λ-coalescent has been introduced independently by Pitman [10] and Sagitov [11].In this process, several blocs may merge at once, but at most one such coalescing event may occur at a given time.
For any t ≥ 0, we denote by N (t) the number of blocs in Π(t).We have in particular N (0) = +∞.We say that the Λ-coalescent comes down from infinity if almost surely N (t) < +∞ for any t > 0. Pitman [10] proved that if Λ({1}) = 0, either the Λ-coalescent comes down from infinity, or N (t) = +∞ for any t > 0 a.s.In the rest of the article, we always assume that Λ has no atom at 1.
Schweinsberg [12] obtained a necessary and sufficient condition for the Λcoalescent to come down from infinity, that Bertoin and Le Gall [3] proved equivalent to Berestycki, Berestycki and Limic [1] obtained the almost sure behaviour for the number of blocs N (t) as t goes to 0, which they called the speed of coming down from infinity.More precisely, setting v ψ (t) = inf{s > 0 : +∞ s dq ψ(q) ≤ t}, they proved that for a Λ-coalescent that comes down from infinity, In this article, we consider the one parameter family of coalescent processes called Beta-coalescents.For any α ∈ (0, 2), we consider the Λ-coalescent such that the measure Λ is Beta(2 − α, α), i.e.
The Beta-coalescents have a number of interesting properties (see e.g.[4,2] and references therein).In particular, if α ∈ (1, 2), it can be constructed as the genealogy of an α-stable continuous state branching process.We observe that thanks to (1.1), α ∈ (1, 2) is a necessary and sufficient condition for the Beta-coalescent to come down from infinity.Moreover, (1.2) can be restated as lim The speed of coming down from infinity for the Beta coalescent can also be found in [2].The main result of this article is a central limit theorem for the number of blocs, as t → 0. Theorem 1.1.Let α ∈ (1, 2) we set (Π(t), t ≥ 0) the Beta(2 − α, α)-coalescent and N (t) = #Π(t) the number of blocs at time t, we have Note that a more precise functional central limit theorem has been obtained by [9] for any Λ-coalescent with a regularly varying density in a neighbourhood of 0. However, our proof follows from simple coupling arguments, that might be of independent interest.Remark 1.2.We observe that the random variable X defined in Theorem 1.1 is an α-stable random variable, that satisfies In Section 2, we use [2] to couple the Beta-coalescent with a stable continuous state branching process, and link the small times behaviour of the number of blocs with the small times behaviour of the continuous-state branching process.In Section 3, we use the so-called Lamperti transform to transfer the computations into the small times asymptotic of an α-stable Lévy process, and use scaling properties to conclude.

Continuous state branching process
A continous-state branching process (or CSBP for short) is a càdlàg (rightcontinuous with left limits at each point) Markov process (Z(t), t ≥ 0) on R + that satisfies the so-called branching property: For any x, y ≥ 0, if (Z x (t), t ≥ 0) and (Z y (t), t ≥ 0) are two independent versions of Z starting from x and y respectively, then the process (Z x (t)+Z y (t), t ≥ 0) is also a version of Z starting from x + y.
The study of CSBP started with the seminal work of [6].As observed in [8,13], there exists a deep connexion between CSBP and Lévy processes.In effect, we observe that for any x, t, λ ≥ 0, the Laplace transform of the CSBP Z satisfies where u is the solution of the following differential equation and φ is the Lévy-Khinchine exponent of a spectrally positive Lévy process (i.e. a Lévy process with no negative jump).The function φ is called the branching mechanism of the CSBP.If φ : λ → λ α with α ∈ (1, 2), we call Z the α-stable CSBP.

Lemma 2.1 ([2]
).For any t > 0, we have N (t) Using this result, to compute the small times behaviour of N (t), it is enough to study the asymptotic behaviour of D(r) and R −1 (t) separately.We first provide a straightforward estimate on the asymptotic behaviour of D. Theorem 2.2.For any α ∈ (1, 2), for any ǫ > 0, we have Proof.We note that (D(r), r > 0) is decreasing.Moreover, for any r ≥ 0, D(r) is a Poisson random variable with parameter θ r = ((α−1)r) − 1 α−1 , by Lemma 2.2 of [2].Therefore, by a deterministic change of variables, it is enough to observe that for any increasing process (P (t), t ≥ 0) such that P (t) is a Poisson random variable with parameter t, we have Using the exponential Markov inequality, for any λ > 0 we have Applying this inequality with λ = t −1/2 , there exists C ǫ > 0 such that for any t ≥ 1, P(P (t) − t > t 1 2 +ǫ ) ≤ C ǫ e −t ǫ .With similar computations, we have We apply the Borel-Cantelli lemma, yielding lim sup n→+∞ As P is increasing, we obtain that for any ǫ > 0, lim t→+∞ = 0 a.s.concluding the proof.

The Lamperti transform
The connexion between CSBP and spectrally positive Lévy processes observed in (2.1) can be strengthen.In [8], Lamperti observed that a CSBP with branching mechanism φ could be constructed as a random time change of a Lévy process with Lévy-Khinchine exponent φ.A proof of this result can be found in [5].More precisely, let (Y (t), t ≥ 0) be a spectrally positive Lévy process starting from a, such that E(e −λY (t) ) = e −aλ+tφ(λ) .We set T = inf{s ≥ 0 : Y (s) ≤ 0} and The Lamperti transform states that for Z a CSBP with branching mechanism φ such that Z(0) = a, we have In the rest of the section, we denote by (Y (t), t ≥ 0) a Lévy process with Lévy-Khinchine exponent φ(λ) = λ α such that Y (0) = 1 a.s.We also set Y 0 (t) = Y (t) − 1.We write T = inf {s ≥ 0 : Y (s) ≤ 0} and Using (3.1), the process defined in (2.3) satisfies Therefore, up to a slight abuse of notation, we write by change of variable, and again R −1 (t) = inf {s ≥ 0 : R(s) ≥ t}.We first prove a central limit theorem for the asymptotic behaviour of R(t) as t → 0.
Theorem 3.1.We denote by X = 1 0 Y 0 (s)ds.We have Proof.For any ǫ > 0 and t > 0, we write A t,ǫ = {|Y (s) − 1| ≤ ǫ, s ≤ 2t} the event such that Y stays in an ǫ neighbourhood of 1 until time 2t.As observed in [2, Lemma 4.2], there exists C > 0 such that P(A c t,ǫ ) ≤ Ctǫ −α .We first prove that lim t→0 U(t) t = 1 and lim t→0 R(t) t = C α a.s.Let ǫ < 1/2, observe that on the event A t,ǫ , we have T > 2t, therefore for any s ≤ t, we have In particular, letting t → 0 we obtain Letting ǫ → 0, this yields lim t→0 U(s) s = 1 a.s.Similarly, by (3.3) we have 1 As a consequence, we have where ds.Note that as Y 0 is an α-stable Lévy process, the following scaling property holds for any λ > 0: We first prove that lim t→0 for any x ∈ (0, 1).Therefore, on the event A t,ǫ , for any s ≤ t, we have Using (3.5) with λ = t, for any δ > 0, we have Letting t → 0, we have lim t→0 t −1− 1 α ∆(t) = 0 in probability.We now study the asymptotic behaviour of t −1− 1 α U(t) 0 Y 0 (s)ds.First observe that for any δ, η > 0, we have As a straightforward consequence of Theorem 3.1, we obtain the asymptotic behaviour of R −1 at small times.