Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-27T22:20:27.923Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems for the minimal position in a branching random walk with independent logconcave displacements

Part of: Stochastic processes Limit theorems Markov processes

Published online by Cambridge University Press:  01 July 2016

Markus Bachmann
Markus Bachmann*
Affiliation:
Purdue University
*
∗ Postal address: Neuhofstr. 17, 60318 Frankfurt, Germany. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider a branching random walk in which each particle has a random number (one or more) of offspring particles that are displaced independently of each other according to a logconcave density. Under mild additional assumptions, we obtain the following results: the minimal position in the nth generation, adjusted by its α-quantile, converges weakly to a non-degenerate limiting distribution. There also exists a ‘conditional limit’ of the adjusted minimal position, which has a (Gumbel) extreme value distribution delayed by a random time-lag. Consequently, the unconditional limiting distribution is a mixture of extreme value distributions.

Keywords

Branching random walkminimal positionage-dependent branching processfirst birth timenonlinear integral operatortravelling waveextreme value distribution

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F05: Central limit and other weak theorems 60G70: Extreme value theory; extremal processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 1 , March 2000 , pp. 159 - 176
Copyright
Copyright © Applied Probability Trust 2000 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Biggins, J. D. (1976). The first- and last-birth problems for a multitype age-dependent branching process. Adv. Appl. Prob. 8, 446–459.Google Scholar
[2] Biggins, J. D. (1977). Chernoff's theorem in the branching random walk. J. Appl. Prob. 14, 630–636.Google Scholar
[3] Biggins, J. D. (1977). Martingale convergence in the branching random walk. J. Appl. Prob. 14, 25–37.CrossRefGoogle Scholar
[4] Biggins, J. D. (1997). How fast does a general branching random walk spread? In Classical and Modern Branching Processes, eds Athreya, K. B. and Jagers, P., IMA Volumes in Mathematics and its Applications, Vol. 84. Springer, New York, pp. 19–39.Google Scholar
[5] Bramson, M. D. (1978). Minimal displacement of branching walk. Z. Wahrscheinlichkeitsth. 45, 89–108.Google Scholar
[6] Bramson, M. D. (1978). Maximal displacement of branching Brownian motion. Commun. Pure Appl. Math. 31, 531–581.Google Scholar
[7] Bramson, M. D. (1983). Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Amer. Math. Soc. 44, 1–190.Google Scholar
[8] Crump, K. S. and Mode, C. J. (1968). A general age-dependent branching process, I. J. Math. Anal. Appl. 24, 494–508.Google Scholar
[9] Dekking, F. M. and Host, B. (1991). Limit distributions for minimal displacement of branching random walks. Prob. Theory Rel. Fields 90, 403–426.CrossRefGoogle Scholar
[10] Dekking, F. M. and Speer, E. R. (1997). On the shape of the wavefront of branching random walk. In Classical and Modern Branching Processes, eds Athreya, K. B. and Jagers, P., IMA Volumes in Mathematics and its Applications, Vol. 84. Springer, New York, pp. 73–88.Google Scholar
[11] Doney, R. A. (1972). A limit theorem for a class of supercritical branching processes. J. Appl. Prob. 9, 707–724.CrossRefGoogle Scholar
[12] Doney, R. A. (1973). On a functional equation for general branching processes. J. Appl. Prob. 10, 198–205.Google Scholar
[13] Durrett, R. (1983). Maxima of branching random walks. Z. Wahrscheinlichkeitsth. 62, 165–170.Google Scholar
[14] Durrett, R. (1996). Probability: Theory and Examples. 2nd edn, Wadsworth, Belmont, CA.Google Scholar
[15] Grill, K. (1996). The range of simple branching random walk. Statist. Prob. Lett. 26, 213–218.Google Scholar
[16] Hammersley, J. M. (1974). Postulates for subadditive processes. Ann. Prob. 2, 652–680.Google Scholar
[17] Joffe, A., Le Cam, L. and Neveu, J. (1973). Sur la loi des grands nombres pour des variables aléatoires de Bernoulli attachées à un arbre dyadique. C. R. Acad. Sci. Paris, Série A 277, 963–964.Google Scholar
[18] Kingman, J. F. C. (1975). The first birth problem for an age-dependent branching process. Ann. Prob. 3, 790–801.Google Scholar
[19] Kolmogorov, A., Petrovsky, I. and Piscounov, N. (1937). Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. Moscow Univ. Math. Bull. 1, 1–25.Google Scholar
[20] Lalley, S. and Sellke, T. (1987). A conditional limit theorem for the frontier of a branching Brownian motion. Ann. Prob. 15, 1052–1061.CrossRefGoogle Scholar
[21] Lalley, S. and Sellke, T. (1992). Limit theorems for the frontier of a one-dimensional branching diffusion. Ann. Prob. 20, 1310–1340.CrossRefGoogle Scholar
[22] Lui, R. (1982). A nonlinear integral operator arising from a model in population genetics I. Monotone initial data. SIAM J. Math. Anal. 13, 913–937.Google Scholar
[23] Lyons, R. (1997). A simple path to Biggins' martingale convergence for branching random walk. In Classical and Modern Branching Processes, eds Athreya, K. B. and Jagers, P., Springer, New York, pp. 217–221.Google Scholar
[24] McDiarmid, C. (1995). Minimal positions in a branching random walk. Ann. Appl. Prob. 5, 128–139.Google Scholar
[25] McKean, H. P. (1975). Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov. Commun. Pure Appl. Math. 28, 323–331.CrossRefGoogle Scholar
[26] Neveu, J. (1987). Multiplicative martingales for spatial branching processes. In Seminar on Stochastic Processes, eds Çinlar, E., Chung, K. L. and Getoor, R. K.. Birkhäuser, Basel, pp. 223–242.Google Scholar
[27] Rockafellar, R. T. (1970). Convex Analysis. Princeton University Press, Princeton, NJ.CrossRefGoogle Scholar