The Annals of Applied Probability

The large deviations of a multi-allele Wright-Fisher process mapped on the sphere

F. Papangelou

Full-text: Open access

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.

Article information

Source
Ann. Appl. Probab., Volume 10, Number 4 (2000), 1259-1273.

Dates
First available in Project Euclid: 22 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1019487616

Digital Object Identifier
doi:10.1214/aoap/1019487616

Mathematical Reviews number (MathSciNet)
MR1810874

Zentralblatt MATH identifier
1073.60511

Subjects
Primary: 60F10: Large deviations
Secondary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]

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

Citation

Papangelou, F. The large deviations of a multi-allele Wright-Fisher process mapped on the sphere. Ann. Appl. Probab. 10 (2000), no. 4, 1259--1273. doi:10.1214/aoap/1019487616. https://projecteuclid.org/euclid.aoap/1019487616


Export citation

References

  • [1] Amari, S. (1985). Differential-Geometrical Methods in Statistics.Lecture Notes in Statist. 28. Springer, Berlin.
  • [2] Atkinson, C. and Mitchell, A. F. (1981). Rao's distance measure. Sankhya Ser.A 43 345-365.
  • [3] Azencott, R. (1980). Grandes D´eviations et Applications.Lecture Notes in Math. 774 1-176. Springer, Berlin.
  • [4] Dawson, D. A. and Feng, S. (1998). Large deviations for the Fleming-Viot process with neutral mutation and selection. Stochastic Processes Appl. 77 207-232.
  • [5] Ewens, W. J. (1979). Mathematical Population Genetics. Springer, New York.
  • [6] Fisher, R. A. (1922). On the dominance ratio. Proc.Roy.Soc.Edinburgh 42 321-341.
  • [7] Fleming, W. H. and Viot, M. (1979). Some measure-valued Markov processes in population genetics theory. Indiana Univ.Math.J. 28 817-843.
  • [8] Gelfand, I. M. and Fomin, S. V. (1963). Calculus of Variations. Prentice-Hall, Englewood Cliffs, NJ.
  • [9] Kimura, M. (1964). Diffusion Models in Population Genetics. Methuen, London.
  • [10] Morrow, G. J. (1992). Large deviation results for a class of Markov chains with applications to an infinite alleles model of population genetics. Ann.Appl.Probab.2 857-905.
  • [11] Papangelou, F. (1996). Large deviations of the Wright-Fisher process. Lecture Notes in Statist. 114 245-252. Springer, New York.
  • [12] Papangelou, F. (1998). Tracing the path of a Wright-Fisher process with one-way mutation in the case of a large deviation. In Stochastic Processes and Related Topics-A Volume in Memory of Stamatis Cambanis (I. Karatzas, B. Rajput and M. S. Taqqu, eds.) 315-330. Birkhauser, Boston.
  • [13] Papangelou, F. (1998). Elliptic and other functions in the large deviations behavior of the Wright-Fisher process. Ann.Appl.Probab.8 182-192.
  • [14] Papangelou, F. (2000). A note on the probability of rapid extinction of alleles in a Wright- Fisher process. In Probability and Statistical Models with Applications-A Volume in Honor of T.Cacoullos (Ch. A. Charalambides, M. V. Koutras and N. Balakrishnan, eds.) 147-154. Chapman and Hall, New York.
  • [15] Schied, A. (1997). Geometric aspects of Fleming-Viot and Dawson-Watanabe processes. Ann.Probab.25 1160-1179.
  • [16] Wentzell, A. D. (1990). Limit Theorems on Large Deviations for Markov Stochastic Processes. Kluwer, Dordrecht.