Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
intertwining of Markov processes; Wright–Fisher diffusion; pure birth process; time of absorption; coupling
intertwining of Markov processes; Wright–Fisher diffusion; pure birth process; time of absorption; coupling
Summary:
It is known that the time until a birth and death process reaches a certain level is distributed as a sum of independent exponential random variables. Diaconis, Miclo and Swart gave a probabilistic proof of this fact by coupling the birth and death process with a pure birth process such that the two processes reach the given level at the same time. Their coupling is of a special type called intertwining of Markov processes. We apply this technique to couple the Wright-Fisher diffusion with reflection at $0$ and a pure birth process. We show that in our coupling the time of absorption of the diffusion is a. s. equal to the time of explosion of the pure birth process. The coupling also allows us to interpret the diffusion as being initially reluctant to get absorbed, but later getting more and more compelled to get absorbed.
References:
[1] Diaconis, P., Fill, J. A.: Strong stationary times via a new form of duality. Ann. Probab. 18 (1990), 4, 1483-1522. DOI 10.1214/aop/1176990628 | MR 1071805
[2] Diaconis, P., Miclo, L.: On times to quasi-stationarity for birth and death processes. J. Theoret. Probab. 22 (2009), 3, 558-586. DOI 10.1007/s10959-009-0234-6 | MR 2530103
[3] Ethier, S. N., Kurtz, T. G.: Markov Processes: Characterization and Convergence. John Wiley and Sons, 1986. DOI 10.1002/9780470316658 | MR 0838085
[4] Fill, J. A.: Strong stationary duality for continuous-time markov chains. Part I: Theory. J. Theoret. Probab. 5 (1992), 1, 45-70. DOI 10.1007/bf01046778 | MR 1144727
[5] Fill, J. A., Lyzinski, V.: Strong stationary duality for diffusion processes. J. Theoret. Probab. 29 (2016), 4, 1298-1338. DOI 10.1007/s10959-015-0612-1 | MR 3571247
[6] Hudec, T.: Absorption Cascades of One-dimensional Diffusions. Master's Thesis, Charles University in Prague, 2016.
[7] Karlin, S., McGregor, J.: Coincidence properties of birth and death processes. Pacific J. Math. 9 (1959), 4, 1109-1140. DOI 10.2140/pjm.1959.9.1109 | MR 0114247
[8] Kent, J. T.: The spectral decomposition of a diffusion hitting time. Ann. Probab. 10 (1082), 1, 207-219. DOI 10.1214/aop/1176993924 | MR 0637387
[9] Liggett, T. M.: Continuous Time Markov Processes: An Introduction. American Mathematical Soc., 2010. DOI 10.1090/gsm/113 | MR 2574430
[10] Mandl, P.: Analytical Treatment of One-dimensional Markov Processes. Springer, 1968. MR 0247667
[11] Rogers, L. C. G., Pitman, J. W.: Markov functions. Ann. Probab. 9 (1981), 4, 573-582. DOI 10.1214/aop/1176994363 | MR 0624684
[12] Swart, J. M.: Intertwining of birth-and-death processes. Kybernetika 47 (2011), 1, 1-14. MR 2807860
Search
Partner of
