Abstract
We consider the branching treeT(n) of the first (n+1) generations of a critical branching process, conditioned on survival till time βn for some fixed β>0 or on extinction occurring at timek n withk n /n→β. We attach to each vertexv of this tree a random variableX(v) and define\(S(v) = \Sigma _{w \varepsilon \pi (0,v)} X(w)\), where π(0,v) is the unique path in the family tree from its root tov. FinallyM n is the maximal displacement of the branching random walkS(·), that isM n =max{S(v):v∈T(n)}. We show that if theX(v), v∈T(n), are i.i.d. with mean 0, then under some further moment conditionn −1/2 M n converges in distribution. In particular {n −1/2 M n } n⩾1 is a tight family. This is closely related to recent results about Aldous' continuum tree and Le Gall's Brownian snake.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aldous, D. (1991). The continuum random tree II: an overview, inStochastic Analysis, London Math. Soc. Lecture Note Series (M. T. Barlow and N. H. Bingham, eds.),167, 23–70.
Aldous, D. (1993a). The continuum random tree III,Ann. Prob. 21, 248–289.
Aldous, D. (1993b). Tree-based models for random distribution of mass,J. Stat. Phys. 73, 625–641.
Athreya, K. B. and Ney, P. E. (1972).Branching Processes, Springer-Verlag.
Billingsley, P. (1968).Convergence of Probability Measures, John Wiley & Sons.
Bramson, M. D. (1978). Maximal displacement for branching Brownian motion,Comm. Pure and Appl. Math. 31, 531–581.
Burkholder, D. L. (1966). Martingale transforms,Ann. Math. Statist. 37, 1494–1504.
Dembo, A., and Zeitouni, O. (1993). Large deviations for random distributions of mass, preprint.
Durrett, R. (1978). Conditional limit theorems for some null recurrent Markov processes,Ann. Prob. 6, 798–818.
Durrett, R., Kesten, H., and Waymire, E. (1991). On weighted heights of random trees,J. Theor. Prob. 4, 223–237.
Dynkin, E. B., and Kuznetsov, S. (1994). Markov snakes and superprocesses, preprint.
Harris, T. E. (1956). First passage and recurrence distributions,Trans. Amer. Math. Soc. 73, 471–486.
Harris, T. E. (1963).The Theory of Branching Processes, Springer-Verlag and Prentice Hall.
Jagers, P. (1975).Branching Processes with Biological Applications, John Wiley & Sons.
Kesten, H. (1986). Subdiffusive behavior of random walk on a random cluster,Ann. Inst. H. Poincaré 22, 425–487.
Kesten, H. (1994). A limit theorem for weighted branching process trees, inThe Dynkin Festschrift. Markov Processes and their Applications (M. Freidlin, ed.), Birkhäuser, pp. 153–166.
Kesten, H., and Lawler, G. F. (1992). A necessary and sufficient condition for making money from fair games,Ann. Prob. 20, 855–882.
Kesten, H., and Pittel, B. (1994). A local limit theorem for the number of nodes, the height and the number of final leaves in a critical branching process tree, preprint.
Kolchin, V. F. (1986). Random Mappings,Optimization Software.
Lee, T-Y. (1990). Conditioned limit theorems of stopped critical branching Bessel processes,Ann. Prob. 18, 272–289.
Le Gall, J-F. (1991). Brownian excursions, trees and measure valued branching processes,Ann. Prob. 19, 1399–1439.
Le Gall, J-F. (1993). A class of path-valued Markov processes and its applications to superprocesses,Prob. Th. Rel. Fields 95, 25–46.
Marcinkiewicz, J., and Zygmund, A. (1938). Quelques théorèmes sur les fonctions indépendantes,Studia Math. 7, 104–120.
Sawyer, S., and Fleischman, J. (1979). Maximum geographic range of a mutant allele considered as a subtype of a Brownian branching random field,Proc. Nat. Acad. Sci., USA 76, 872–875.
Rights and permissions
About this article
Cite this article
Kesten, H. Branching random walk with a critical branching part. J Theor Probab 8, 921–962 (1995). https://doi.org/10.1007/BF02410118
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02410118