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Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
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Cited by 46 (3 self)
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Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
Asymptotic results on the length of coalescent trees
 Ann. Appl. Prob
"... Abstract. We give the asymptotic distribution of the length of partial coalescent trees for Beta and related coalescents. This allows us to give the asymptotic distribution of the number of (neutral) mutations in the partial tree. This is a first step to study the asymptotic distribution of a natura ..."
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Cited by 23 (2 self)
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Abstract. We give the asymptotic distribution of the length of partial coalescent trees for Beta and related coalescents. This allows us to give the asymptotic distribution of the number of (neutral) mutations in the partial tree. This is a first step to study the asymptotic distribution of a natural estimator of DNA mutation rate for species with large families. 1.
The asymptotic distribution of the length of betacoalescent trees
 Ann. Appl. Probab
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Regeneration in Random Combinatorial Structures
, 2009
"... Theory of Kingman’s partition structures has two culminating points • the general paintbox representation, relating finite partitions to hypothetical infinite populations via a natural sampling procedure, • a central example of the theory: the EwensPitman twoparameter partitions. In these notes we ..."
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Cited by 7 (2 self)
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Theory of Kingman’s partition structures has two culminating points • the general paintbox representation, relating finite partitions to hypothetical infinite populations via a natural sampling procedure, • a central example of the theory: the EwensPitman twoparameter partitions. In these notes we further develop the theory by • passing to structures enriched by the order on the collection of categories, • extending the class of tractable models by exploring the idea of regeneration, • analysing regenerative properties of the EwensPitman partitions, • studying asymptotic features of the regenerative compositions.
On the length of an external branch in the betacoalescents
, 2012
"... In this paper, we consider Beta(2 − α,α) (with 1 < α < 2) and related Λcoalescents. If T (n) denotes the length of an external branch of the ncoalescent, we prove the convergence of n α−1 T (n) when n tends to ∞, and give the limit. To this aim, we give asymptotics for the number σ (n) of ..."
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Cited by 4 (3 self)
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In this paper, we consider Beta(2 − α,α) (with 1 < α < 2) and related Λcoalescents. If T (n) denotes the length of an external branch of the ncoalescent, we prove the convergence of n α−1 T (n) when n tends to ∞, and give the limit. To this aim, we give asymptotics for the number σ (n) of collisions which occur in the ncoalescent until the end of the chosen external branch, and for the block counting process associated with the ncoalescent.
Convergence of some leader election algorithms
, 2008
"... We start with a set of n players. With some probability P(n, k), we kill n −k players; the other ones stay alive, and we repeat with them. What is the distribution of the number Xn of phases (or rounds) before getting only one player? We present a probabilistic analysis of this algorithm under some ..."
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Cited by 4 (2 self)
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We start with a set of n players. With some probability P(n, k), we kill n −k players; the other ones stay alive, and we repeat with them. What is the distribution of the number Xn of phases (or rounds) before getting only one player? We present a probabilistic analysis of this algorithm under some conditions on the probability distributions P(n, k), including stochastic monotonicity and the assumption that roughly a fixed proportion α of the players survive in each round. We prove a kind of convergence in distribution for Xn − log 1/α n; as in many other similar problems there are oscillations and no true limit distribution, but suitable subsequences converge, and there is an absolutely continuous random variable Z such that d(Xn, ⌈Z+log 1/α n⌉) → 0, where d is either the total variation distance or the Wasserstein distance. Applications of the general result include the leader election algorithm where players are eliminated by independent coin tosses and a variation of the leader election algorithm proposed by W.R. Franklin [7]. We study the latter algorithm further, including numerical
ASYMPOTIC BEHAVIOR OF THE TOTAL LENGTH OF EXTERNAL BRANCHES FOR BETACOALESCENTS
"... Abstract. In this paper, we consider the Beta(2 − α,α)coalescents with 1 < α < 2 and study the moments of external branches, in particular the total external branch length L (n) ext of an initial sample of n individuals. For this class of coalescents, it has been proved that n α−1 (n) (d) T → ..."
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Abstract. In this paper, we consider the Beta(2 − α,α)coalescents with 1 < α < 2 and study the moments of external branches, in particular the total external branch length L (n) ext of an initial sample of n individuals. For this class of coalescents, it has been proved that n α−1 (n) (d) T → T, where T (n) is the length of an external branch chosen at random, and T is a known non negative random variable. We get the asymptotic behaviour of several moments of L (n) ext. As a consequence, we obtain that for Beta(2−α,α)coalescents with 1 < α < 2, lim n→+ ∞ n3α−5 E[(L (n) ext −n2−α E[T]) 2] =
On a random recursion related to absorption times of death Markov chains
, 2007
"... Let X1, X2,... be a sequence of random variables satisfying the d distributional recursion X1 = 0 and Xn = Xn−In + 1 for n = 2, 3,..., where In is a random variable with values in {1,..., n − 1} which is independent of X2,..., Xn−1. The random variable Xn can be interpreted as the absorption time of ..."
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Cited by 3 (1 self)
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Let X1, X2,... be a sequence of random variables satisfying the d distributional recursion X1 = 0 and Xn = Xn−In + 1 for n = 2, 3,..., where In is a random variable with values in {1,..., n − 1} which is independent of X2,..., Xn−1. The random variable Xn can be interpreted as the absorption time of a suitable death Markov chain with state space N: = {1, 2,...} and absorbing state 1, conditioned that the chain starts in the initial state n. This paper focuses on the asymptotics of Xn as n tends to infinity under the particular but important assumption that the distribution of In satisfies P{In = k} = pk/(p1+ · · ·+pn−1) for some given probability distribution pk = P{ξ = k}, k ∈ N. Depending on the tail behaviour of the distribution of ξ, several scalings for Xn and corresponding limiting distributions come into play, among them stable distributions and distributions of exponential integrals of subordinators. The methods used in this paper are mainly probabilistic. The key tool is a coupling technique which relates the distribution of Xn to a random walk, which explains, for example, the appearance of the MittagLeffler distribution in this context.