Dynamics of the time to the most recent common ancestor in a large branching population

Annals of Applied Probability

Volumn 20, Issue 1, 2010, Pages 1-25

  Evans, Steven N ^a   Ralph, Peter L ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Bessel; Continuous state branching; Duality; Genealogy; Most recent common ancestor; MRCA; Piecewise deterministic; Poisson cut out; Poisson point process; Recurrence; Self similar; Stationary distribution; Transience

Indexed keywords

EID: 76449094191     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/09-AAP616     Document Type: Article

References (34)

1. BOXMA, O., PERRY, D., STADJE, W. and ZACKS, S. (2006 ). A Markovian growth-collapse model. Adv. in Appi. Probab. 38 221-243. MR2213972
2. BOROVKOV, K. and VERE-JONES, D. (2000). Explicit formulae for stationary distributions of stress release processes. J. Appi. Probab. 37 315-321. MR1780992
3. COSTA, O.L.V. and DUFOUR, F. (2008). Stability and ergodicity of piecewise deterministic Markov processes. SIAM J. Control Optim. 47 1053-1077. MR2385873
4. COLOMBO, G. and DAI PRA, P. (2001). A class of piecewise deterministic Markov processes. Markov Process. Related Fields 7 251-287. MR1856497
5. DAVIS, M. H. A. (1984). Piecewise-deterministic Markov processes: A general class of non- diffusion stochastic models. J. Roy. Statist. Soc. Ser. B 46 353-388. MR790622
6. DAVIS, M. H. A. (1993). Markov Models and Optimization. Monographs on Statistics and Applied Probability 49 Chapman and Hall, London. MR1283589
7. DUFOUR, F. and COSTA, O L. V. (1999). Stability of piecewise-deterministic Markov processes. SIAM J. Control Optim. 37 1483-1502. MR1710229
8. DUMAS, V., GUILLEMIN, F and ROBERT, P. (2002). A Markovian analysis of additive-increase multiplicative-decrease algorithms. Adv. in Appl. Probab. 34 85-111. MR1895332
9. DALEY, D J. and VERE-JONES, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York. MR950166
10. ELIAZAR, I. (2007). The M/G/∞ system revisited: Finiteness, summability, long range dependence, and reverse engineering. Queueing Syst. 55 71-82. MR2293566
11. EVANS, S.N and PERKINS, E. (1990). Measure-valued Markov branching processes conditioned on nonextinction. Israel J. Math. 71 329-337. MR1088825
12. EVANS, S.N and PITMAN, J. (1998). Stationary Markov processes related to stable Ornstein- Uhlenbeck processes and the additive coalescent. Stochastic Process. Appl. 77 175-185. MR1649003
13. EVANS, S.N., PITMAN, J and WINTER, A. (2006). Rayleigh processes, real trees, and root growth with re-grafting. Probab. Theory Related Fields 134 81-126. MR2221786
14. EVANS, S. N. (1992). The entrance space of a measure-valued Markov branching process conditioned on nonextinction. Canad. Math. Bull. 35 70-74. MR1157466
15. EVANS, S. N. (1993). Two representations of a conditioned superprocess. Proc. Roy. Soc. Edinburgh Sect. A 123 959-971. MR1249698
16. ETHERIDGE, A. M. and WILLIAMS, D. R. E. (2003). A decomposition of the (1 + β)- superprocess conditioned on survival. Proc. Roy. Soc. Edinburgh Sect. A 133 829-847. MR2006204
17. FELLER, W. (1951). Diffusion processes in genetics. In Proc. Second Berkeley Symp. Math. Statist. Probab. 227-246. Univ. California Press, Berkeley. MR0046022
18. FlTZSIMMONS, P. J., FRISTEDT, B. and SHEPP, L. A. (1985). The set of real numbers left uncovered by random covering intervals. Z. Wahrsch. Verw. Gebiete 70 175-189. MR799145
19. INGMAN, M., KAESSMANN, H., PAABO, S and GYLLENSTEN, U. (2000). Mitochondrial genome variation and the origin of modern humans. Nature 408 708-713.
20. JACOD, J and SKOROKHOD, A. V. (1996). Jumping Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 32 11-67. MR1373726
21. KALLENBERG, O (2000). An extension of the basic Palm measure correspondence. Probab. Theory Related Fields 117 113-131. MR1759510
22. KNIGHT, F. B. (1981). Essentials of Brownian Motion and Diffusion. Mathematical Surveys 18 Amer. Math. Soc., Providence, RI. MR613983
23. MANDELBROT, B. B. (1972). Renewal sets and random cutouts. Z. Wahrsch. Verw. Gebiete 22 145-157. MR0309162
24. MORGAN, F. (2009). Geometric Measure Theory: A Beginner's Guide, 4th ed. Elsevier/Academic Press, Amsterdam. MR2455580
25. MEYN, S.P and TWEEDIE, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London. MR1287609
26. PFAFFELHUBER, P and WAKOLBINGER, A. (2006). The process of most recent common ancestors in an evolving coalescent. Stochastic Process. Appl. 116 1836-1859. MR2307061
27. PITMAN, J. and YOR, M. (1981). Bessel processes and infinitely divisible laws. In Stochastic Integrals (Proc. Sympos., Univ. Durham, Durham, 1980). Lecture Notes in Math. 851 285-370. Springer, Berlin. MR620995
28. PITMAN, J. and YOR, M. (1982). A decomposition of Bessel bridges. Z. Wahrsch. Verw. Gebiete 59 425-157. MR656509
29. SlMON, D. and DERRIDA, B. (2006). Evolution of the most recent common ancestor of a population with no selection. J. Stat. Mech. Theory Exp. P05002.
30. SHEPP, L. A. (1972). Covering the line with random intervals. Z. Wahrsch. Verw. Gebiete 23 163-170. MR0322923
31. TAKÁCS, L. (1962). Introduction to the Theory of Queues. Oxford Univ. Press, New York. MR0133880
32. TRAVERS, S. A. A., CLEWLEY, J. P., GLYNN, J.R., FINE, P.E.M., CRAMPIN, A.C., SIBANDE, F., MULAWA, D., MCINERNEY, J O. and MCCORMACK, G. P. (2004). Timing and reconstruction of the most recent common ancestor of the subtype C clade of Human Immunodeficiency Virus type 1. J. Virol. 78 10501-10506.
33. THOMSON, R., PRITCHARD, J K., SHEN, P., OEFNER, P.J and FELDMAN, M. W. (2000). Recent common ancestry of human Y chromosomes: Evidence from DNA sequence data. Proc. Nat. Acad. Sci. India Sect. A 97 7360-7365.
34. VOLKMAN, S.K., BARRY, A.E., LYONS, E.J., NIELSEN, K. M., THOMAS, S.M., MEHEE, C., THAKORE, S.S., DAY, K.P., WIRTH, D. F. and HARTL, D. L. (2001). Recent origin of Plasmodium falciparum from a single progenitor. Science 293 482-484.