Previous |  Up |  Next


intertwining of Markov processes; birth and death process; averaged Markov process; first passage time; coupling; eigenvalues
It has been known for a long time that for birth-and-death processes started in zero the first passage time of a given level is distributed as a sum of independent exponentially distributed random variables, the parameters of which are the negatives of the eigenvalues of the stopped process. Recently, Diaconis and Miclo have given a probabilistic proof of this fact by constructing a coupling between a general birth-and-death process and a process whose birth rates are the negatives of the eigenvalues, ordered from high to low, and whose death rates are zero, in such a way that the latter process is always ahead of the former, and both arrive at the same time at the given level. In this note, we extend their methods by constructing a third process, whose birth rates are the negatives of the eigenvalues ordered from low to high and whose death rates are zero, which always lags behind the original process and also arrives at the same time.
[1] Athreya, S. R., Swart, J. M.: Survival of contact processes on the hierarchical group. Probab. Theory Related Fields 147 (2010), 3, 529–563. DOI 10.1007/s00440-009-0214-x | MR 2639714 | Zbl 1191.82028
[2] Diaconis, P., Miclo, L.: On times to quasi-stationarity for birth and death processes. J. Theor. Probab. 22 (2009), 558–586. DOI 10.1007/s10959-009-0234-6 | MR 2530103 | Zbl 1186.60086
[3] Fill, J. A.: Strong stationary duality for continuous-time Markov chains. I. Theory. J. Theor. Probab. 5 (1992), 1, 45–70. DOI 10.1007/BF01046778 | MR 1144727 | Zbl 0746.60075
[4] HASH(0x2434598). F. R. Gantmacher: The Theory of Matrices, Vol. 2. AMS, Providence 2000.
[5] Karlin, S., McGregor, J.: Coincidence properties of birth and death processes. Pac. J. Math. 9 (1959), 1109–1140. DOI 10.2140/pjm.1959.9.1109 | MR 0114247 | Zbl 0097.34102
[6] Miclo, L.: On absorbing times and Dirichlet eigenvalues. ESAIM Probab. Stat. 14 (2010), 117–150. DOI 10.1051/ps:2008037 | MR 2654550
[7] Rogers, L. C. G., Pitman, J. W.: Markov functions. Ann. Probab. 9 (1981), 4, 573–582. MR 0624684 | Zbl 0466.60070
Partner of
EuDML logo