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An unexpected connection between branching processes and optimal stopping

  • David Assaf (a1), Larry Goldstein (a2) and Ester Samuel-Cahn (a1)


A curious connection exists between the theory of optimal stopping for independent random variables, and branching processes. In particular, for the branching process Z n with offspring distribution Y, there exists a random variable X such that the probability P(Z n = 0) of extinction of the nth generation in the branching process equals the value obtained by optimally stopping the sequence X 1,…, X n , where these variables are i.i.d. distributed as X. Generalizations to the inhomogeneous and infinite horizon cases are also considered. This correspondence furnishes a simple ‘stopping rule’ method for computing various characteristics of branching processes, including rates of convergence of the nth generation's extinction probability to the eventual extinction probability, for the supercritical, critical and subcritical Galton-Watson process. Examples, bounds, further generalizations and a connection to classical prophet inequalities are presented. Throughout, the aim is to show how this unexpected connection can be used to translate methods from one area of applied probability to another, rather than to provide the most general results.


Corresponding author

Postal address: Department of Statistics, Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel
∗∗ Postal address: Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA
∗∗∗ Email address:


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Athreya, K. B., and Ney, P. E. (1972). Branching Processes. Springer, New York.
Chow, Y., Robbins, H. and Siegmund, D. (1971). Great Expectations: The Theory of Optimal Stopping, Houghton Mifflin, Boston.
D'Souza, J. C. (1995). The extinction time of the inhomogeneous branching process. In Branching Processes, ed. Heyde, C. C. (Lecture Notes in Statist. 99). Springer, Berlin, pp. 106117.
De Haan, L. (1976). Sample extremes: an elementary introduction. Statist. Neerlandica 30, 161172.
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.
Hill, T. P., and Kertz, R. P. (1981). Ratio comparisons of supremum and stop rule expectations. Z. Wahrscheinlichkeitsth. 56, 283285.
Hill, T. P., and Kertz, R. P. (1982). Comparisons of stop rule and supremum expectations of i.i.d. random variables. Ann. Prob. 10, 336345.
Jagers, P. (1974). Galton–Watson processes in varying environments. J. Appl. Prob. 11, 174178.
Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, London.
Jirina, M. (1976). Extinction of non-homogeneous Galton–Watson processes. J. Appl. Prob. 13, 132137.
Karlin, S., and Taylor, H. M. (1975). A First Course in Stochastic Processes. Second edn., Academic Press, New York.
Keiding, N., and Nielsen, J. E. (1975). Branching processes with varying and geometric offspring distribution. J. Appl. Prob. 12, 135141.
Kennedy, D. P., and Kertz, R. P. (1991). The asymptotic behavior of the reward sequence in the optimal stopping of i.i.d. random variables. Ann. Prob. 9, 329341.
Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
Slack, R. S. (1968). A branching process with mean one and possibly infinite variance. Z. Wahrscheinlichkeitsth. 9, 139145.


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