Some large deviations in Kingman's coalescent

Abstract

Kingman's coalescent is a random tree that arises from classical population genetic models such as the Moran model. The individuals alive in these models correspond to the leaves in the tree and the following two laws of large numbers concerning the structure of the tree-top are well-known: (i) The (shortest) distance, denoted by Tn, from the tree-top to the level when there are n lines in the tree satisfies nTn n→∞ − −−− → 2 almost surely; (ii) At time Tn, the population is naturally partitioned in exactly n families where individuals belong to the same family if they have a common ancestor at time Tn in the past. If Fi,n denotes the size of the ith family, then n(F 2 1,n + · · · + F 2 n,n ) n→∞ − −−− → 2 almost surely. For both laws of large numbers we prove corresponding large deviations results. For (i), the rate of the large deviations is n and we can give the rate function explicitly. For (ii), the rate is n for downwards deviations and √ n for upwards deviations. For both cases we give the exact rate function.

The Kingman tree length process has infinite quadratic variation

“…Moreover, the integrability condition [0,1] h 2 ν(dh) < ∞, which must be satisfied by a Lévy measure, is violated by the jump intensity measure of L ld . Indeed, the expectation of the life-length T G of a line born at level k is 2/k (see (3) below) and for large k the distribution of T G is concentrated around 2/k (see the proof of Theorem 1, which uses a result of [4]). Since the points of η k come at rate k − 1, the jump intensity measure of L ld has (for large k) mass k − 1 concentrated around 2/k.…”

“…Cramér's theorem guarantees that P M k / ∈ 1 2 (k − 1)(t − s), 2(k − 1)(t − s) decays exponentially in k and hence the first term on the right-hand side is summable. For the second term we use Theorem 1 of [4] which says that the sequence (kT k ) k≥2 (that converges a.s. to 2 as k → ∞) satisfies a large deviation principle with scale k and a good rate function. Since…”

“…where d G is the death time of line G. The jump size of the process L ld,N at time d G,N has size equal to the life-length T G,N of the line G up to level N defined in (4). Note that the exponential times X G j do not depend on N .…”

Limit theorems for pure death processes coming down from infinity

“…Here we prove only the statement (i), the statement (ii) is proved similarly. Our proof of (i) is more direct than that of [4] and uses the classical Cramer's device of 'tilted distributions'. Let x > 1.…”

“…In Section 3 we give a number of examples illustrating a wide range of possible growth patterns covered by Theorem 2 for the speed function v(t) as t → 0. Section 4 presents an explicit large deviation theorem generalizing a recent result in [4] obtained for the Kingman coalescent. The remaining sections are devoted to self-contained proofs.…”