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November 2000 The large deviations of a multi-allele Wright-Fisher process mapped on the sphere
F. Papangelou
Ann. Appl. Probab. 10(4): 1259-1273 (November 2000). DOI: 10.1214/aoap/1019487616

Abstract

This is the fourth in a series of papers devoted to the study of the large deviations of a Wright –Fisher process modeling the genetic evolution of a reproducing population.Variational considerations imply that if the process undergoes a large deviation, then it necessarily follows closely a definite path from its original to its current state. The favored paths were determined previously for a one-dimensional process subject to one-way mutation or natural selection, respectively, acting on a faster time scale than random genetic drift. The present paper deals with a general $d$-dimensional Wright–Fisher process in which any mutation or selection forces act on a time scale no faster than that of genetic drift. If the states of the process are represented as points on a $d$-sphere, then it can be shown that the position of a subcritically scaled process at a fixed “time” $T$ satisfies a large-deviation principle with rate function proportional to the square of the length of the great circle arc joining this position with the initial one (Hellinger–Bhattacharya distance). If a large deviation does occur, then the process follows with near certainty this arc at constant speed. The main technical problem circumvented is the degeneracy of the covariance matrix of the process at the boundary of the state space.

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F. Papangelou. "The large deviations of a multi-allele Wright-Fisher process mapped on the sphere." Ann. Appl. Probab. 10 (4) 1259 - 1273, November 2000. https://doi.org/10.1214/aoap/1019487616

Information

Published: November 2000
First available in Project Euclid: 22 April 2002

zbMATH: 1073.60511
MathSciNet: MR1810874
Digital Object Identifier: 10.1214/aoap/1019487616

Subjects:
Primary: 60F10
Secondary: 60J20

Keywords: Hellinger-Bhattacharya distance , large deviations , mutation , natural selection , random genetic drift , Rate function , Wright-Fisher process

Rights: Copyright © 2000 Institute of Mathematical Statistics

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Vol.10 • No. 4 • November 2000
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