Abstract
We consider a branching random walk for which the maximum position of a particle in the n’th generation, R n , has zero speed on the linear scale: R n /n → 0 as n → ∞. We further remove (“kill”) any particle whose displacement is negative, together with its entire descendence. The size Z of the set of un-killed particles is almost surely finite (Gantert and Müller in Markov Process. Relat. Fields 12:805–814, 2006; Hu and Shi in Ann. Probab. 37(2):742–789, 2009). In this paper, we confirm a conjecture of Aldous (Algorithmica 22:388–412, 1998; and Power laws and killed branching random walks) that E [Z] < ∞ while \({{\mathbf E}\left[Z\,{\rm log}\, Z\right]=\infty}\) . The proofs rely on precise large deviations estimates and ballot theorem-style results for the sample paths of random walks.
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Addario-Berry L., Reed B.: Minima in branching random walks. Ann. Probab. 37(3), 1044–1079 (2009)
Addario-Berry, L., Reed, B.: Ballot theorems for random walks with finite variance. arXiv:0802.2491 [math.PR] (2008)
Aïdékon, E.: Tail asymptotics for the total progeny of the critical killed branching random walk. arXiv:0911.0877 [math.PR] (2009)
Aldous D.: A Metropolis-type optimization algorithm on the infinite tree. Algorithmica 22, 388–412 (1998)
Aldous, D.: Power laws and killed branching random walks. http://www.stat.berkeley.edu/~aldous/Research/OP/brw.html
Aldous D.J.: Greedy search on the binary tree with random edge-weights. Comb. Probab. Comput. 1, 281–293 (1992)
Alon N., Spencer J.: The Probabilitic Method, 3rd edn. Wiley, New York (2008)
Athreya K.B., Ney P.E.: Branching Processes. Springer, Berlin (1972)
Bahadur R.R., Ranga Rao R.: On deviations of the sample mean. Ann. Math. Stat. 31, 1015–1027 (1960)
Berry A.C.: The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Am. Math. Soc. 49, 122–136 (1941)
Bertrand J.: Solution d’un problème. C. R. Acad. Sci. Paris 105, 369 (1887)
Biggins J.D.: Chernoff’s theorem in the branching random walk. J. Appl. Probab. 14, 630–636 (1977)
Biggins J.D., Kyprianou A.E.: Measure change in multitype branching. Adv. Appl. Probab. 36, 544–581 (2004)
Chauvin B., Rouault A.: KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees. Probab. Theory Relat. Fields 80, 299–314 (1988)
Chernoff H.: A measure of the asymptotic efficiency for tests of a hypothesis based on the sum of observables. Ann. Math. Stat. 2, 493–509 (1952)
Chung K.L., Erdős P.: On the application of the Borel–Cantelli lemma. Trans. Am. Math. Soc. 72, 179–186 (1952)
Dembo A., Zeitouni O.: Large Deviation Techniques and Applications, 2nd edn. Springer, New York (1998)
Derrida, B., Simon, D.: The survival probability of a branching random walk in presence of an absorbing wall. EPL 78(60006) (2007)
Donsker M.D.: Justification and extension of Doob’s heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Stat. 23, 277–281 (1952)
Esséen C.G.: Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law. Acta Math. 77, 1–125 (1963)
Fekete M.:Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten. Math. Z. 17, 228–249 (1923)
Feller W.: An Introduction to Probability Theory and its Applications, vol. I, 3rd edn. Wiley, New York (1968)
Feller W.: An Introduction to Probability Theory and its Applications, vol. II, 3rd edn. Wiley, New York (1971)
Flajolet P., Odlyzko A.: Singularity analysis of generating functions. SIAM J. Discrete Math. 3, 216–240 (1990)
Flajolet P., Sedgewick R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)
Fortuin C.M., Kasteleyn P.W., Ginibre J.: Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22, 89–103 (1971)
Gantert N., Müller S.: The critical branching Markov chain is transient. Markov Process. Relat. Fields 12, 805–814 (2006)
Gantert, N., Hu, Y., Shi, Z.: Asymptotics for the survival probability in a killed branching random walk. arXiv:0811.0262 [math.PR] (2009)
Hammersley J.M.: Postulates for subadditive processes. Ann. Probab. 2, 652–680 (1974)
Harris J.W., Harris S.C.: Survival probabilities for branching Brownian motion with absorption. Electron. Commun. Probab. 12, 81–92 (2007)
Harris T.E.: A lower bound for the critical probability in a certain percolation process. Proc. Camb. Philos. Soc. 56, 13–20 (1960)
Hu Y., Shi Z.: Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37(2), 742–789 (2009)
Kahane J.P., Peyrière J.: Sur certaines martingales de Benoît Mandelbrot. Adv. Math. 22, 131–145 (1976)
Karp R.M., Pearl J.: Searching for an optimal path in a tree with random costs. Artif. Intell. 21, 99–116 (1983)
Kesten H.: Branching Brownian motion with absorption. Stoch. Process. Appl. 7, 9–47 (1978)
Kesten H., Stigum B.P.: Additional limit theorems for indecomposable multidimensional Galton–Watson processes. Ann. Math. Stat. 37, 1463–1481 (1966)
Kingman J.F.C.: The first birth problem for an age-dependent branching process. Ann. Probab. 3, 790–801 (1975)
Le Gall J.-F.: Random trees and applications. Probab. Surv. 2, 245–311 (2005)
Lyons R.: A simple path to Biggins’ martingale convergence for branching random walk. In: Athreya, K.B., Jagers, P. (eds) Classical and Modern Branching Processes, pp. 217–222. Springer, New York (1997)
Lyons R., Pemantle R., Peres Y.: Conceptual proofs of the L log L criteria for mean behavior of branching processes. Ann. Probab. 23, 1125–1138 (1995)
Maillard, P.: The number of absorbed individuals in branching Brownian motion with a barrier. arXiv:1004.1426v1 [math.PR] (2010)
Neveu J.: Arbres et processus de Galton–Watson. Annales de l’I. H. P. Probabilités et statistiques 22, 199–207 (1986)
Pemantle R.: Search cost for a nearly optimal path in a binary tree. Ann. Appl. Probab. 19, 1273–1291 (2009)
Pemantle, R.: Critical killed branching process tail probabilities. Manuscript (1999)
Pemantle R., Peres Y.: Critical random walk in random environment on trees. Ann. Probab. 23(1), 105–140 (1995)
Petrov V.V.: On the probabilities of large deviations for sums of independent random variables. Theory Probab. Appl. 10, 287–298 (1965)
Petrov, V.V.: Sums of independent random variables. In: A Series of Modern Surveys in Mathematics, vol. 82. Springer-Verlag (1975)
Revuz D., Yor M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (2004)
Rogers L.C.G., Williams D.: Diffusions, Markov Processes and Martingales: Itô Calculus, vol. 2, 2nd edn. Cambridge University Press, Cambridge (2000)
Simon D., Derrida B.: Quasi-stationary regime of a branching random walk in presence of an absorbing wall. J. Stat. Phys. 131, 203–233 (2008)
Steele, J.M.: Probability theory and combinatorial optimization. In: CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1997)
Stone, C.J.: On local and ratio limit theorems. In: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, pp. 217–224 (1965)
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Addario-Berry, L., Broutin, N. Total progeny in killed branching random walk. Probab. Theory Relat. Fields 151, 265–295 (2011). https://doi.org/10.1007/s00440-010-0299-2
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DOI: https://doi.org/10.1007/s00440-010-0299-2