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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.
The structure of the allelic partition of the total population for Galton-Watson processes with neutral mutations
"... We consider a (sub-)critical Galton–Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We specify the law of this allelic partition in terms of the distribution of the number of clone-children and ..."
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Cited by 29 (4 self)
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We consider a (sub-)critical Galton–Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We specify the law of this allelic partition in terms of the distribution of the number of clone-children and the number of mutant-children of a typical individual. The approach combines an extension of Harris representation of Galton–Watson processes and a version of the ballot theorem. Some limit theorems related to the distribution of the allelic partition are also given. 1. Introduction. We consider a Galton–Watson process, that is, a population model with asexual reproduction such that at every generation, each individual gives birth to a random number of children according to a fixed distribution and independently of the other individuals in the population. We are interested in the situation where a child can be either a clone, that
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 allelic partition for coalescent point processes
, 2008
"... Assume that individuals alive at time t in some population can be ranked in such a way that the coalescence times between consecutive individuals are i.i.d. The ranked sequence of these branches is called a coalescent point process. We have shown in a previous work [14] that splitting trees are impo ..."
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Cited by 15 (6 self)
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Assume that individuals alive at time t in some population can be ranked in such a way that the coalescence times between consecutive individuals are i.i.d. The ranked sequence of these branches is called a coalescent point process. We have shown in a previous work [14] that splitting trees are important instances of such populations. Here, individuals are given DNA sequences, and for a sample of n DNA sequences belonging to distinct individuals, we consider the number Sn of polymorphic sites (sites at which at least two sequences differ), and the number An of distinct haplotypes (sequences differing at one site at least). It is standard to assume that mutations arrive at constant rate (on germ lines), and never hit the same site on the DNA sequence. We study the mutation pattern associated with coalescent point processes under this assumption. Here, Sn and An grow linearly as n grows, with explicit rate. However, when the branch lengths have infinite expectation, Sn grows more rapidly, e.g. as n ln(n) for critical birth–death processes. Then, we study the frequency spectrum of the sample, that is, the numbers of polymorphic sites/haplotypes carried by k individuals in the sample. These numbers are shown to grow also linearly with sample size, and we provide simple explicit formulae for mutation frequencies and haplotype frequencies. For critical birth–death processes, mutation frequencies are given by the harmonic series and haplotype frequencies by Fisher’s logarithmic series. Running head. The allelic partition for coalescent point processes.
The number of small blocks in exchangeable random partitions
, 2010
"... Suppose Π is an exchangeable random partition of the positive integers and Πn is its restriction to {1,...,n}. Let Kn denote the number of blocks of Πn, and let Kn,r denote the number of blocks of Πn containing r integers. We show that if 0 < α < 1 and Kn/(n α ℓ(n)) converges in probability t ..."
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Cited by 4 (0 self)
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Suppose Π is an exchangeable random partition of the positive integers and Πn is its restriction to {1,...,n}. Let Kn denote the number of blocks of Πn, and let Kn,r denote the number of blocks of Πn containing r integers. We show that if 0 < α < 1 and Kn/(n α ℓ(n)) converges in probability to Γ(1 − α), where ℓ is a slowly varying function, then Kn,r/(n α ℓ(n)) converges in probability to αΓ(r − α)/r!. This result was previously known when the convergence of Kn/(n α ℓ(n)) holds almost surely, but the result under the hypothesis of convergence in probability has significant implications for coalescent theory. We also show that a related conjecture for the case when Kn grows only slightly slower than n fails to be true.
Asymptotic sampling formulae for Λ-coalescents
, 2012
"... 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 Λ-coalescent. This allows us to derive an exa ..."
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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 Λ-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 ∞. Some of our results hold in the case of a general Λ-coalescent that comes down from infinity, but we obtain more precise information under a regular variation assumption. In this case, we obtain results of independent interest for the time at which a mutation uniformly chosen at random was generated. This exhibits a phase transition at α = 3/2, where α ∈ (1, 2) is the exponent of regular variation. AMS 2000 Subject Classification. 60J25, 60F99, 92D25 Key words and phrases. Λ-coalescents, speed of coming down from infinity, exchangeable
ON THE NUMBER OF ALLELIC TYPES FOR SAMPLES TAKEN FROM EXCHANGEABLE COALESCENTS WITH MUTATION
, 2008
"... Let Kn denote the number of types of a sample of size n taken from an exchangeable coalescent process (Ξ-coalescent) with mutation. A distributional recursion for the sequence (Kn)n∈N is derived. If the coalescent does not have proper frequencies, i.e., if the characterizing measure Ξ on the infinit ..."
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Cited by 2 (1 self)
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Let Kn denote the number of types of a sample of size n taken from an exchangeable coalescent process (Ξ-coalescent) with mutation. A distributional recursion for the sequence (Kn)n∈N is derived. If the coalescent does not have proper frequencies, i.e., if the characterizing measure Ξ on the infinite simplex ∆ does not have mass at zero and satisfies R ∆ |x|Ξ(dx)/(x, x) < ∞, where |x |: = P∞ i=1 xi and (x,x): = P∞
Almost sure asymptotics for the number of types for simple Ξ-coalescents
- ELECTRONIC COMMUNICATIONS IN PROBABILITY
, 2012
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