Skip to main content Accessibility help

Convergence to stationarity in the Moran model

  • Peter Donnelly (a1) and Eliane R. Rodrigues (a2)


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 A 1 and A 2. 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 A 1. 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(-γec ) 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.


Corresponding author

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:


Hide All
Aldous, D., and Diaconis, P. (1986). Shuffling cards and stopping times. Am. Math. Monthly 93, 333348.
Diaconis, P. (1988). Group Representation in Probability and Statistics. Lecture Notes 11. IMS Monogr. Ser., ed. Gupta, Shanti S., Institute of Mathematical Statistics, Hayward.
Diaconis, P. (1996). The cutoff phenomenon in finite Markov chains. Proc. Natl. Acad. Sci. USA 36, 16591664.
Diaconis, P., and Fill, J. A. (1990). Strong stationary times via a new form of duality. Ann. Prob. 18, 14831522.
Diaconis, P., and Saloff-Coste, L. (1994). Moderate growth and random walks on finite groups. Geom. Functional Anal. 4, 136.
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. 4475.
Diaconis, P., and Saloff-Coste, L. (1996). Nash inequalities for finite Markov chains. J. Theoret. Prob. 9, 459510.
Diaconis, P., and Shahshahani, M. (1987). Time to stationarity in the Bernoulli-Laplace diffusion model. SIAM J. Math. Anal. 18, 208218.
Diaconis, P., and Stroock, D. (1991). Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Prob. 1, 3661.
Donnelly, P. (1984). Transient behaviour of the Moran model in population genetics. Math. Proc. Camb. Phil. Soc. 95, 349358.
Donnelly, P. (1986). Dual processes in population genetics. Stochastic Spatial Processes (Lecture Notes in Math. 1212). Springer, Berlin, pp. 95105.
Donnelly, P., Lloyd, P., and Sudbury, A. (1994). Approach to stationarity of the Bernoulli–Laplace diffusion model. Adv. Appl. Prob. 26, 715727.
Donnelly, P. and Tavaré, S. (1986). The ages of alleles and a coalescent. Adv. Appl. Prob. 18, 119.
Fill, J. A. (1992). Strong stationary duality for continuous-time Markov chains. Part I: theory. J. Theoret. Prob. 5, 4570.
Griffiths, R. C. (1980). Lines of descent in the diffusion approximation of neutral Wright–Fisher models. Theoret. Popn Biol. 17, 3750.
Karlin, S., and McGregor, J. (1962). On a genetic model by Moran. Proc. Camb. Phil. Soc. 58, 299311.
Karlin, S., and Rinott, Y. (1980). Classes of ordering of measures and related correlation inequalities I. Multivariate totally positive distributions. J. Multivariate Anal. 10, 467498.
Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.
Matthews, P. (1992). Strong stationary times and eigenvalues. J. Appl. Prob. 29, 228233.
Moran, P. A. P. (1958). Random processes in genetics. Proc. Camb. Phil. Soc. 54, 6072.
Rosenthal, J. S. (1995). Convergence rates for Markov chains. SIAM Rev. 37, 387405.
Tavaré, S. (1984). Lines-of-descent and genealogical processes, and their applications in population genetics models. Theoret. Popn Biol. 26, 119164.
Watterson, G. A. (1984). Lines of descent and the coalescent. Theoret. Popn Biol. 26, 7792.


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed