Asymptotic expansions and precise deviations in the Kingman coalescent

In this paper, we study the small-time asymptotic behavior of the Kingman coalescent. We obtain the Berry-Esseen bound and the Edgeworth expansion in the central limit theorem. Moreover, by the method of mod-φ convergence, we also obtain the precise large deviations and the precise moderate deviations. Last, we also obtain a nonasymptotic deviation inequality for the Kingman coalescent.


Introduction
The Kingman coalescent was introduced in 1982 by Kingman in [5]. It was used to describe the genealogy of a sample from a population. Ever since its appearance, the Kingman coalescent has found many applications in biology. It also serves as the dual process of the Fleming-Viot process with parent-independent mutation. As a dual process, the Kingman coalescent usually describes the backward movement in time in population genetics. More generally, coalescent processes, the Lambda coalescence in particular, are dual processes of some generalized Fleming-Viot processes.
Apart from describing sample genealogy, the Kingman coalescent also has other interpretations, such as the number of surviving ancient families. Due to genetic drift and mutation, the ancient families are lost as time moves forward. Let D t be the number of surviving ancient families up to time t. Then D t is a pure-death Markov chain with transition rates q n,n−1 = n 2 , q nk = 0, k < n − 1, starting at ∞. Once an ancient family disappears, the Kingman coalescent experiences a coalescing event. These coalescing events arrive independently, and waiting times between coalescing events follow exponential distributions. Let T n be the arriving time of the coalescing event, where n is the number of surviving ancient families right after T n . Then T n = we study the small-time behavior of D t , this dual relation facilitates our computation a lot.
As a standard coalescent model in population genetics, the Kingman coalescent has been thoroughly studied. For instance, the law of large numbers and the central limit theorem were summarized in [2](see also [7] for some discussion on the central limit theorem). Some functional central limit theorem was also obtained in [9]. Large deviation principle has been discussed in [3]. However, large deviation principle and moderate deviation principle only aim at seeking a leading term of the logarithm of tail probability. Therefore, polynomial correction terms will be missing if we only consider large deviations and moderate deviations. Especially, these correction terms play indispensable roles in statistical inference. The precise deviations, however, not only provide the leading terms but also polynomial correction terms of the tail probability. As far as we know, the precise large deviations and precise moderate deviations for the Kingman coalescent are still unknown. Similarly, we have not seen any asymptotic result like the Berry-Esseen bounds, the Edgeworth expansions and deviation inequalities for the Kingman coalescent in the existing literatures either. Similar problems for the Lambda-coalescents also deserve study.
In this paper, we are going to find the Berry-Esseen bound, establish the Edgeworth expansions in the central limit theorem(CLT for short) and the local central limit theorem. Some precise large deviations, precise moderate deviations, and deviation inequalities are also obtained for the Kingman coalescent. Due to the dual relation P(T n ≥ t) = P(D t ≥ n), we will only focus on T n . The asymptotic behavior of D t is only a simple corollary of the asymptotic behavior of T n . The main methodology is an asymptotic analysis of the generating function in the complex domain and the mod-φ convergence theory [4].
Our paper is organized as follows. In Section 2, will state the main results. In section 3, we show the Berry-Esseen bound and the Edgeworth expansions in the CLT. The Edgeworth expansions in local CLT will be presented in Section 4. Precise large deviations and precise moderate deviations are shown in Section 5. In Section 6, we will prove the deviation inequality.

Main results
We will rescale nT n as Z n = √ 3n 2 (nT n − 2). In the following, we are going to provide the main results on the asymptotic behavior of the rescaled quantity Z n . We will mainly discuss the Edgeworth expansion in the CLT. The expansion involves Hermite polynomials A notation S k will be repeatedly used in the Edgeworth expansion. It is a set of k-tuple (m 1 , · · · , m k ), where m 1 , . . . , m k are non-negative integers and 1 · m 1 + 2 · m 2 + · · · + k · m k = k. (2.1) Now we are ready to state the main results.

Local limit theorem
Let p n (x) be the density of Z n . By local limit theorem, we mean the asymptotic behavior of p n (x) in uniform topology as n → ∞. First, we establish the Edgeworth expansion in uniform topology, then the local limit theorem will be a simple corollary.
When we consider the Edgeworth expansion, the cumulative distribution P( , to approximate the density p n (x) of Z n . Theorem 2.4. (Local limit theorem). There exists a constant C > 0 such that  ], by dual relation P(D t ≥ n) = P(T n ≥ t), one can also obtain the CLT the Edgeworth expansion of D t in the CLT and its Edgeworth expansions in the local central limit theorem.

Precise deviations
The strong limit of nT n is 2 as n → ∞(refer to [2]). Thus, for > 0, both {nT n > 2 + } and {nT n < 2 − } are rare events. Precise deviation principle is about the estimations of P(nT n > 2 + ) and P(nT n < 2 − ), where indicates the level of deviations from the strong limit point 2. Large deviations correspond to the case = O(1), and moderate deviations corresponds to another case = o(1).
Precise large deviations and moderate deviations can be obtained by mod-φ convergence [4], which depends on carefully analysis of the moment generating function Ee zn 2 Tn of nT n in complex domain S (−∞,1/2) := {z ∈ C; Re(z) < 1/2}. In section 4, we will show that ψ n (z) = e −nη(z) Ee zn 2 Tn has a uniform limit on S (−∞,1/2) , where Here η(z) can be regarded as the analytic continuation of the limiting log-Laplace transformation η(θ), which has been obtained in [3]. The rate function I(x) is the Legendre transformation of η(θ), θ ∈ R, and

Remark 2.10.
The rescaled quantity Z n has either positive or negative deviation. When the absolute deviation is small, the decay is of Gaussian type; when the absolute deviation is large, the decay behaves like a Poisson tail.

Berry-Esseen bounds and asymptotic expansions of CLT
In this section, we are going to prove the Berry-Esseen bound and the Edgeworth expansions in the CLT. Note that Berry-Esseen bound is only a special case of the Edgeworth expansions in the CLT. Thus, it suffices to show the Edgeworth expansions in the CLT. But first we need the following basic lemma (refer to [8]). Lemma 3.1. Let F be a cumulative distribution function. Assume that G is a differentiable function, satisfying G(−∞) = 0 and G(+∞) = 1, and G is bounded. Then for any r > 0, where φ F (s) := R e isx dF (x) is the Fourier transform of function F .
By Lemma 3.1, we can establish the Edgeworth expansions of cumulative distribution P(Z n ≤ x) of Z n through expansion of its characteristic function φ n (θ) = Ee iθZn . Note is the cumulative distribution function of the standard normal distribution and H l−1 (x) are Hermite polynomials. By integration by parts, we have which builds up a correspondence between expansions of P(Z n ≤ x) and its characteristic function φ n (θ).
Due to the correspondence relation (3.1), we know In the following, we will show that P(Z n ≤ x) has an expansion G m,n (x) = Φ(x) + m−2 k=1 Q k,n (x).
Proof of Theorem 2.2. To apply Lemma 3.1, we take r = Cn (m−1)/2 , where 0 < C < min{ 14/9, δ}. Then There exists a positive constant M , independent of n, such that 24 It remains to show that both K 1 and K 2 are bounded by M n (m−1)/2 . Note that It is easy to see that |Ψ n (θ)| ≤ M 2(m−2) l=0 |θ| l e − θ 2 2 . Then by Gaussian probability estimates, we know there exist two positive constants C 1 , C 2 such that Moreover, By the definition of T n , we have Ee x l l , switching summation order yields φ n (θ) = exp ∞ l=2 Introducing n,2 = A n,2 − 1 3 and w = i 4 2 − n n,2 l = 2.
One can also show that there exists constants N, 0 < δ < 1 8 , such that a l (n) ≤ 2l! + 1, ∀n ≥ N , and ∀ |w| < δ, Next we will expand f (w) = e −g(w) to get the expansion of φ n (θ). We claim that we can expand f (w) as . To specify f (k) (0), we turn to Faá di Bruno formula(see [1] and [6]). To this end, we define h(z) = e −z , then f (w) = h(g(w)). Applying Faá di Bruno formula to f (w), we have Last, we can rewrite φ n (θ) as

Local limit theorems
We only need to prove Theorem 2.5 because Theorem 2.4 is a simple corollary of Theorem 2.5. By the correspondence (3.1), one can deduce that Applying inverse Fourier transform, we know By Lemma 3.2, there exist positive constants C 1 and C 2 such that |θ| s e −θ 2 /6 dθ < ∞.
Meanwhile, in the proof of Theorem 2.2, we have already shown that there exist positive constants D 1 , D 2 such that |I 2 | ≤ D 1 e −D2n . Thus we have proved Theorem 2.5.

Mod-φ convergence and precise deviations
In this section, we will show precise deviation principle. The key tool is the mod-φ convergence theory. First we certify a Lemma on mod-φ convergence for Kingman coalescent.  Proof. By the definition of T n and η(z), we know where log(1 − 2z x 2 ) is the principal branch. It is not difficult to see that ψ n (z) is analytic on S (−∞,1/2) because Re(1 − 2z x 2 ) > 1 − 2Re(z) > 0. Next, for given z ∈ S (−∞,1/2) , we will show that log ψ n (z) = n 1 n ds, (5.4) and 1 n ds.
Meanwhile, the estimation of P Z n ≤ x , x < 0, can be done similarly by applying the above procedure to −Z n .