Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T15:16:25.314Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence to stationarity in the Moran model

Part of: Markov processes Distribution theory

Published online by Cambridge University Press:  14 July 2016

Peter Donnelly  and
Eliane R. Rodrigues
Peter Donnelly*
Affiliation:
University of Oxford
Eliane R. Rodrigues*
Affiliation:
Instituto de Mateméticas - UNAM
*
∗Postal address: Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK
∗∗Postal address: Instituto de Mateméticas - UNAM, Area de la Investigación Científica, Circuito Exterior - Ciudad Universitária, México, DF 04510, México. Email address: eliane@gauss.matem.unam.mx
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider a population of fixed size consisting of N haploid individuals. Assume that this population evolves according to the two-allele neutral Moran model in mathematical genetics. Denote the two alleles by A1 and A2. Allow mutation from one type to another and let 0 < γ < 1 be the sum of mutation probabilities. All the information about the population is recorded by the Markov chain X = (X(t))t≥0 which counts the number of individuals of type A1. In this paper we study the time taken for the population to ‘reach’ stationarity (in the sense of separation and total variation distances) when initially all individuals are of one type. We show that after t∗ = Nγ-1logN + cN the separation distance between the law of X(t∗) and its stationary distribution converges to 1 - exp(-γe-γc) as N → ∞. For the total variation distance an asymptotic upper bound is obtained. The results depend on a particular duality, and couplings, between X and a genealogical process known as the lines of descent process.

Keywords

Rate of convergence to stationaritycut-off phenomenoncouplinggenealogical processes

MSC classification

Primary: 60J99: None of the above, but in this section
Secondary: 92D10: Genetics 62E17: Approximations to distributions (nonasymptotic)
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 3 , September 2000 , pp. 705 - 717
Copyright
Copyright © Applied Probability Trust 2000 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aldous, D., and Diaconis, P. (1986). Shuffling cards and stopping times. Am. Math. Monthly 93, 333–348.CrossRefGoogle Scholar
Diaconis, P. (1988). Group Representation in Probability and Statistics. Lecture Notes 11. IMS Monogr. Ser., ed. Gupta, Shanti S., Institute of Mathematical Statistics, Hayward.Google Scholar
Diaconis, P. (1996). The cutoff phenomenon in finite Markov chains. Proc. Natl. Acad. Sci. USA 36, 1659–1664.Google Scholar
Diaconis, P., and Fill, J. A. (1990). Strong stationary times via a new form of duality. Ann. Prob. 18, 1483–1522.CrossRefGoogle Scholar
Diaconis, P., and Saloff-Coste, L. (1994). Moderate growth and random walks on finite groups. Geom. Functional Anal. 4, 1–36.CrossRefGoogle Scholar
Diaconis, P., and Saloff-Coste, L. (1995). Random walks on finite groups: a survey of analytic techniques. In Probability Measures on Groups and Related Structures, XI (Oberwolfach, 1994), ed. Heyer, H. World Scientific, River Edge, pp. 44–75.Google Scholar
Diaconis, P., and Saloff-Coste, L. (1996). Nash inequalities for finite Markov chains. J. Theoret. Prob. 9, 459–510.Google Scholar
Diaconis, P., and Shahshahani, M. (1987). Time to stationarity in the Bernoulli-Laplace diffusion model. SIAM J. Math. Anal. 18, 208–218.Google Scholar
Diaconis, P., and Stroock, D. (1991). Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Prob. 1, 36–61.CrossRefGoogle Scholar
Donnelly, P. (1984). Transient behaviour of the Moran model in population genetics. Math. Proc. Camb. Phil. Soc. 95, 349–358.CrossRefGoogle Scholar
Donnelly, P. (1986). Dual processes in population genetics. Stochastic Spatial Processes (Lecture Notes in Math. 1212). Springer, Berlin, pp. 95–105.Google Scholar
Donnelly, P., Lloyd, P., and Sudbury, A. (1994). Approach to stationarity of the Bernoulli–Laplace diffusion model. Adv. Appl. Prob. 26, 715–727.Google Scholar
Donnelly, P. and Tavaré, S. (1986). The ages of alleles and a coalescent. Adv. Appl. Prob. 18, 1–19.CrossRefGoogle Scholar
Fill, J. A. (1992). Strong stationary duality for continuous-time Markov chains. Part I: theory. J. Theoret. Prob. 5, 45–70.CrossRefGoogle Scholar
Griffiths, R. C. (1980). Lines of descent in the diffusion approximation of neutral Wright–Fisher models. Theoret. Popn Biol. 17, 37–50.CrossRefGoogle ScholarPubMed
Karlin, S., and McGregor, J. (1962). On a genetic model by Moran. Proc. Camb. Phil. Soc. 58, 299–311.CrossRefGoogle Scholar
Karlin, S., and Rinott, Y. (1980). Classes of ordering of measures and related correlation inequalities I. Multivariate totally positive distributions. J. Multivariate Anal. 10, 467–498.Google Scholar
Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar
Matthews, P. (1992). Strong stationary times and eigenvalues. J. Appl. Prob. 29, 228–233.Google Scholar
Moran, P. A. P. (1958). Random processes in genetics. Proc. Camb. Phil. Soc. 54, 60–72.CrossRefGoogle Scholar
Rosenthal, J. S. (1995). Convergence rates for Markov chains. SIAM Rev. 37, 387–405.CrossRefGoogle Scholar
Tavaré, S. (1984). Lines-of-descent and genealogical processes, and their applications in population genetics models. Theoret. Popn Biol. 26, 119–164.Google Scholar
Watterson, G. A. (1984). Lines of descent and the coalescent. Theoret. Popn Biol. 26, 77–92.CrossRefGoogle Scholar