# The Passage Time Distribution for a Birth-and-Death Chain: Strong Stationary Duality Gives a First Stochastic Proof

Article

## Abstract

A well-known theorem usually attributed to Keilson states that, for an irreducible continuous-time birth-and-death chain on the nonnegative integers and any d, the passage time from state 0 to state d is distributed as a sum of d independent exponential random variables. Until now, no probabilistic proof of the theorem has been known. In this paper we use the theory of strong stationary duality to give a stochastic proof of a similar result for discrete-time birth-and-death chains and geometric random variables, and the continuous-time result (which can also be given a direct stochastic proof) then follows immediately. In both cases we link the parameters of the distributions to eigenvalue information about the chain. We also discuss how the continuous-time result leads to a proof of the Ray–Knight theorem.

Intimately related to the passage-time theorem is a theorem of Fill that any fastest strong stationary time T for an ergodic birth-and-death chain on {0,…,d} in continuous time with generator G, started in state 0, is distributed as a sum of d independent exponential random variables whose rate parameters are the nonzero eigenvalues of −G. Our approach yields the first (sample-path) construction of such a T for which individual such exponentials summing to T can be explicitly identified.

## Keywords

Markov chains Birth-and-death chains Passage time Strong stationary duality Anti-dual Eigenvalues Stochastic monotonicity Ray–Knight theorem

## Mathematics Subject Classification (2000)

60J25 60J35 60J10 60G40

## References

1. 1.
Aldous, D., Diaconis, P.: Strong uniform times and finite random walks. Adv. Appl. Math. 8, 69–97 (1987)
2. 2.
Brown, M., Shao, Y.S.: Identifying coefficients in the spectral representation for first passage time distributions. Probab. Eng. Inform. Sci. 1, 69–74 (1987)
3. 3.
Cox, J.T., Roesler, U.: A duality relation for entrance and exit laws for Markov monotone Markov processes. Ann. Probab. 13, 558–565 (1985)
4. 4.
Diaconis, P., Fill, J.: Strong stationary times via a new form of duality. Ann. Probab. 18, 1483–1522 (1990)
5. 5.
Diaconis, P., Miclo, L.: On times to quasi-stationarity for birth and death processes. J. Theor. Probab. (2009). doi: Google Scholar
6. 6.
Diaconis, P., Saloff-Coste, L.: Separation cut-offs for birth and death chains. Ann. Appl. Probab. 16, 2098–2122 (2006)
7. 7.
Fill, J.A.: Strong stationary duality for continuous-time Markov chains. Part I: Theory. J. Theor. Probab. 5, 45–70 (1992)
8. 8.
Fill, J.A.: An interruptible algorithm for perfect sampling via Markov chains. Ann. Appl. Probab. 8, 131–162 (1998)
9. 9.
Fill, J.A.: On hitting times and fastest strong stationary times for skip-free and more general chains. J. Theor. Probab. (2009). doi: Google Scholar
10. 10.
Karlin, S., McGregor, J.: Coincidence properties of birth and death processes. Pac. J. Math. 9, 1109–1140 (1959)
11. 11.
Keilson, J.: Log-concavity and log-convexity in passage time densities for of diffusion and birth-death processes. J. Appl. Probab. 8, 391–398 (1971)
12. 12.
Keilson, J.: Markov Chain Models—Rarity and Exponentiality. Springer, New York (1979)
13. 13.
Kent, J.T.: The appearance of a multivariate exponential distribution in sojourn times for birth-death and diffusion processes. In: Probability, Statistics and Analysis, London Math. Soc. Lecture Note Ser., vol. 79, pp. 161–179. Cambridge Univ. Press, Cambridge (1983) Google Scholar
14. 14.
Knight, F.B.: Random walks and a sojourn density process of Brownian motion. Trans. Am. Math. Soc. 109, 56–86 (1963)
15. 15.
Matthews, P.: Strong stationary times and eigenvalues. J. Appl. Probab. 29, 228–233 (1992)
16. 16.
Micchelli, C.A., Willoughby, R.A.: On functions which preserve the class of Stieltjes matrices. Linear Algebra Appl. 23, 141–156 (1979)
17. 17.
Ray, D.: Sojourn times of diffusion processes. Ill. J. Math. 7, 615–630 (1963)