A Central Limit theorem for Markov Processes that Move by Small Steps

Norman, M. Frank

doi:10.1214/aop/1176996498

Cited by 41 publications

(15 citation statements)

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“…More detailed statements with the precise conditions on the parameters of the SLFVS are given in Section 2. A very similar result was proved by F. Norman in the non-spatial setting [Nor75a] (see also [Nor74a], [Nor77] and [Nor75b]). Norman considered the Wright-Fisher model for a population of size N under natural selection (see [Eth09] for an introduction to such models).…”

Section: Introductionsupporting

confidence: 78%

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A central limit theorem for the spatial $\Lambda $-Fleming-Viot process with selection

Forien¹,

Penington²

2017

Electron. J. Probab.

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We study the evolution of gene frequencies in a population living in R d , modelled by the spatial Λ-Fleming-Viot process with natural selection ([BEV10], [EVY14]). We suppose that the population is divided into two genetic types, a and A, and consider the proportion of the population which is of type a at each spatial location. If we let both the selection intensity and the fraction of individuals replaced during reproduction events tend to zero, the process can be rescaled so as to converge to the solution to a reaction-diffusion equation (typically the Fisher-KPP equation, [EVY14]). We show that the rescaled fluctuations converge in distribution to the solution to a linear stochastic partial differential equation. Depending on whether offspring dispersal is only local or if large scale extinction-recolonization events are allowed to take place, the limiting equation is either the stochastic heat equation with a linear drift term driven by space-time white noise or the corresponding fractional heat equation driven by a coloured noise which is white in time. If individuals are diploid (i.e. either AA, Aa or aa) and if natural selection favours heterozygous (Aa) individuals, a stable intermediate gene frequency is maintained in the population. We give estimates for the asymptotic effect of random fluctuations around the equilibrium frequency on the local average fitness in the population. In particular, we find that the size of this effect -known as the drift load -depends crucially on the dimension d of the space in which the population evolves, and is reduced relative to the case without spatial structure.AMS 2010 subject classifications. Primary: 60G57 60F05 60J25 92D10. Secondary: 60G15.2. Update q as in (6).Note that if we let w = |B(x, r)| −1 B(x,r) q t − (z) dz, then at a neutral reproduction event, P (k = a) = w and at a selective event, P (k = a) = w − F (w).Remark. The existence of a unique Ξ-valued process following these dynamics under condition (4) is proved in [EVY14, Theorem 1.2] in the special case F (w) = w(1 − w) (in the neutral case s = 0, this was done in [BEV10]). In our general case, the condition on w − F (w) allows us to define a dual process and hence prove existence and uniqueness in the same way as in [EVY14].We shall consider two different distributions µ for the radii of events, i) the fixed radius case : µ(dr) = δ R (dr) for some R > 0, ii) the stable radius case : µ(dr) = 1 r≥1 r d+α+1 dr for a fixed α ∈ (0, 2 ∧ d).

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Section: Introductionsupporting

confidence: 78%

“…Robertson's result can be made rigorous using tools found in [Nor74a] and [Nor74b]. We adapt these to our setting and study the same effect in spatially structured populations.…”

Section: Introductionmentioning

confidence: 99%

A central limit theorem for the spatial $\Lambda $-Fleming-Viot process with selection

Forien¹,

Penington²

2017

Electron. J. Probab.

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“…We also remark that these results are different from the celebrated law of large numbers for Markov chains established by Kurtz (1971Kurtz ( , 1981, where the convergence is established for sample paths of the CTMCs over a finite time interval or for a sequence of t M that increases M increases (Norman 1974), not for the stationary distributions of the CTMCs. The contributions of our results are two-fold: First, our results provide a direct method of studying the convergence of stationary distributions of stochastic systems to their mean-field limits.…”

Section: Introductioncontrasting

confidence: 72%

On the Approximation Error of Mean-Field Models

Lü

2018

Stochastic Systems

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Abstract. Mean-field models have been used to study large-scale and complex stochastic systems, such as large-scale data centers and dense wireless networks, using simple deterministic models (dynamical systems). This paper analyzes the approximation error of mean-field models for continuous-time Markov chains (CTMCs) and focuses on mean-field models that are represented as finite-dimensional dynamical systems with a unique equilibrium point. By applying Stein's method and the perturbation theory, the paper shows that under some mild conditions, if the mean-field model is globally asymptotically stable and locally exponentially stable, the mean-square difference between the stationary distribution of the stochastic system of size M (i.e., with M nodes/particles) and the equilibrium point of the corresponding mean-field system is O(1/M). The result of this paper establishes a general theorem for establishing the convergence and the approximation error (i.e., the rate of convergence) of a large class of CTMCs to their mean-field limits by mainly looking into the stability of the mean-field model, which is a deterministic system and is often easier to analyze than the CTMCs. Two applications of mean-field models in data center networks are presented to demonstrate the novelty of our results.

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“…Near these boundary values, drift determines the dynamics. Hence, a standard approach is to treat p t as a deterministic process when it is in the range < p < 1 − for 1 (Kurtz 1971, Norman 1974, Kaplan et al 1989, Stephan et al 1992. Once the allele reaches frequency 1 − , it is assumed to be quickly fixed by drift and this additional time is assumed small and ignored.…”

Section: C)mentioning

confidence: 99%

Hitchhiking and Selective Sweeps

Walsh¹,

Lynch²

2018

Oxford Scholarship Online

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When a favorable allele increases in frequency, it alters the coalescent structure (the pattern of times back to a common ancestor) at linked sites relative to that under drift. This creates patterns of sequence polymorphism than can be used to potentially detect ongoing, or very recent, selection. This idea of a neutral allele hitchhiking up to high frequency when coupled to a favorable allele is the notion of a selective sweep, and this chapter reviews the considerable body of associated population-genetics theory on sweeps. Different types of sweeps leave different signatures, resulting in the very diverse collection of tests of selection discussed in Chapter 9. Either a history of recurrent sweeps, or of background selection, results in linked genomic regions of reduced effective population size. This implies that more mutations in sich regions are efficiently neutral, which can result in increased substitution rates and lower codon bias. Finally, the chapter examines the theory for when response is expected to start from existing variation, as opposed to waiting for the appearance of new mutations.

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A Central Limit theorem for Markov Processes that Move by Small Steps

Cited by 41 publications

References 0 publications

A central limit theorem for the spatial $\Lambda $-Fleming-Viot process with selection

A central limit theorem for the spatial $\Lambda $-Fleming-Viot process with selection

On the Approximation Error of Mean-Field Models

Hitchhiking and Selective Sweeps

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