Population genetics models with skewed fertilities: A forward and backward analysis

Stochastic Models

Volumn 27, Issue 3, 2011, Pages 521-554

  Huillet, Thierry ^a   Möhle, Martin ^b  

^a UNIVERSITÉ DE CERGY PONTOISE   (France)

^b UNIVERSITY OF TÜBINGEN   (Germany)

Author keywords

Ancestral process; Cannings model; Compound Poisson model; Dirichlet model; Dirichlet Kingman coalescent; Duality; Evolutionary processes; Exchangeable coalescent; Karlin and McGregor model; Kimura model; Kingman coalescent; Population dynamics; Simultaneous multiple collisions; Wright Fisher model

Indexed keywords

ANCESTRAL PROCESS; COMPOUND POISSON; DIRICHLET MODEL; DIRICHLET-KINGMAN COALESCENT; DUALITY; EVOLUTIONARY PROCESS; EXCHANGEABLE COALESCENT; KARLIN AND MCGREGOR MODEL; KIMURA MODEL; KINGMAN COALESCENT; MULTIPLE COLLISIONS; WRIGHT-FISHER MODELS;

POPULATION STATISTICS;

CHROMOSOMES;

EID: 80051531736     PISSN: 15326349     EISSN: 15324214     Source Type: Journal    
DOI: 10.1080/15326349.2011.593411     Document Type: Article

References (34)

1. Berestycki, N.; Pitman, J. Gibbs distributions for random partitions generated by a fragmentation process. Journal of Statistical Physics 2007, 127, 381-418.
2. Caballero, A. Developments in the prediction of effective population size. Heredity 1994, 73, 657-679.
3. Cannings, C. The latent roots of certain Markov chains arising in genetics: a new approach. I. Haploid models. Advances in Applied Probability 1974, 6, 260-290.
4. Cannings, C. The latent roots of certain Markov chains arising in genetics: a new approach. II. Further haploid models. Advances in Applied Probability 1975, 7, 264-282.
5. Comtet, L. Analyse Combinatoire; Volumes 1 and 2. Presses Universitaires de France: Paris, 1970.
6. Eldon, B.; Wakeley, J. Coalescent processes when the distribution of offspring number of individuals is highly skewed. Genetics 2006, 172, 2621-2633.
7. Ethier, S.N.; Kurtz, T.G. Markov Processes. Characterization and Convergence. Wiley: New-York, 1986.
8. Ewens, W.J. The sampling theory of selectively neutral alleles. Theoretical Population Biology 1972, 3, 87-112.
9. Gladstien, K. The characteristic values and vectors for a class of stochastic matrices arising in genetics. SIAM Journal of Applied Mathematics 1978, 34, 630-642.
10. Handa, K. The two-parameter Poisson-Dirichlet point process. Bernoulli 2010, 15, 1082-1116.
11. Huillet, T.; Martinez, S. Dirichlet-Kingman partition revisited. Far East Journal of Theoretical Statistics 2008, 24, 1-33.
12. Karlin, S.; McGregor, J. Direct product branching processes and related Markov chains. Proceeding of the National Academy & Sciences USA 1964, 51, 598-602.
13. Karlin, S.; McGregor, J. Direct product branching processes and related Markov chains. I. Calculations of rates of approach to homozygosity. Proceeding of the International Research Seminar. Springer: Berlin, pp. 111-145, 1965.
14. Kingman, J.F.C. The coalescent. Stochastic Process and their Applications 1982, 13, 235-248.
15. Kolchin, V.F. Random Mappings. Translation Series in Mathematics and Engineering. Optimization Software Inc. Publications Division, New York, 1986
16. Liggett, T.M. Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer-Verlag: New York, 1985.
17. Maruyama, T. Stochastic Problems in Population Genetics. Lecture Notes in Biomathematics 17. Springer-Verlag: Berlin-New York, 1977.
18. McSweeney, J.K.; Pittel, B.G. Expected coalescence time for a nonuniform allocation process. Advances in Applied Probability 2008, 40, 1002-1032.
19. Möhle, M. Robustness results for the coalescent. Journal of Applied Probability 1998, 35, 438-447.
20. Möhle, M. The concept of duality and applications to Markov processes arising in neutral population genetics models. Bernoulli 1999, 5, 761-777.
21. Möhle, M. Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models. Advances in Applied Probability 2000, 32, 983-993. (Pubitemid 32869493)
22. Möhle, M. Asymptotic results for coalescent processes without proper frequencies and applications to the two-parameter Poisson-Dirichlet coalescent. Stochastic Processes and their Applications 2010, 120, 2159-2173.
23. Möhle, M. Looking forwards and backwards in the multi-allelic neutral Cannings population model. Journal of Applied Probability 2010, 47, 713-731.
24. Möhle, M.; Sagitov, S. A classification of coalescent processes for haploid exchangeable population models. Annals of Probability 2001, 29, 1547-1562. (Pubitemid 33406000)
25. Möhle, M.; Sagitov, S. Coalescent patterns in diploid exchangeable population models. Journal of Mathematical Biology. 2003, 47, 337-352. (Pubitemid 38261039)
26. Moran, P.A.P.; Watterson, G.A. The genetic effects of family structure in natural populations. Australian Journal of Biological Sciences 1959, 12, 1-15.
27. Pitman, E.J.G. On the behaviour of the characteristic function of a probability distribution in the neighbourhood of the origin. Journal of the Australian Mathematical Society 1968, 8, 422-443.
28. Pitman, J. Combinatorial Stochastic Processes. Lecture Notes in Mathematics 1875, 13-35, 2006.
29. Sagitov, S. Convergence to the coalescent with simultaneous multiple mergers. Journal of Applied Probability 2003, 40, 839-854. (Pubitemid 38191715)
30. Sagitov, S.; Jagers, P. The coalescent effective size of age-structured populations. Annals of Applied Probability 2005, 15, 1778-1797. (Pubitemid 43205545)
31. Schweinsberg, J. Coalescents with simultaneous multiple collisions. Electronic Journal of Probability 2000, 5, Paper no. 12, 50 pp.
32. Schweinsberg, J. Coalescent processes obtained from supercritical Galton-Watson processes. Stochastic Processes and their Applications 2003, 106, 107-139. (Pubitemid 36613991)
33. Steutel, F.W.; van Harn, K. Discrete analogues of self-decomposability and stability. Annals of Probability 1979, 7, 893-899.
34. Wakeley, J.; Sargsyan, O. Extensions of the coalescent effective population size. Genetics 2009, 181, 341-345.