Seneta-Heyde norming in the branching random walk. (English) Zbl 0873.60062

Summary: In the discrete-time supercritical branching random walk, there is a Kesten-Stigum type result for the martingales formed by the Laplace transform of the \(n\)th generation positions. Roughly, this says that for suitable values of the argument of the Laplace transform the martingales converge in mean provided an “\(X\log X\)” condition holds. Here it is established that when this moment condition fails, so that the martingale converges to zero, it is possible to find a (Seneta-Heyde) renormalization of the martingale that converges (in probability) to a finite nonzero limit when the process survives. As part of the proof, a Seneta-Heyde renormalization of the general (Crump-Mode-Jagers) branching process is obtained; in this case the convergence holds almost surely. The results rely heavily on a detailed study of the functional equation that the Laplace transform of the limit must satisfy.


60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G50 Sums of independent random variables; random walks
Full Text: DOI


[1] ASMUSSEN, S. and HERING, H. 1983. Branching Processes. Birkhauser, Boston. \" · Zbl 0516.60095
[2] BIGGINS, J. D. 1977a. Martingale convergence in the branching random walk. J. Appl. Probab. 14 25 37. · Zbl 0997.60093
[3] BIGGINS, J. D. 1977b. Chernoff’s Theorem in the branching random walk. J. Appl. Probab. 14 630 636. · Zbl 0997.60093
[4] BIGGINS, J. D. 1992. Uniform convergence of martingales in the branching random walk. Ann. Probab. 20 137 151. · Zbl 0748.60080
[5] BIGGINS, J. D. and KYPRIANOU, A. E. 1996. Branching random walk: Seneta Heyde norming. In Z Trees: Proceedings of a Workshop, Versailles June 14 16, 1995 B. Chauvin, S. Cohen. and A. Rouault, eds.. Birkhauser, Basel. \" · Zbl 0864.60070
[6] CHAUVIN, B. 1988. Arbres et Processus de Branchement. Ph.D. thesis, Univ. Paris 6.
[7] CHAUVIN, B. 1991. Product martingales and stopping lines for branching Brownian motion. Ann. Probab. 19 1195 1205. · Zbl 0738.60079
[8] CHAUVIN, B. and ROUAULT, A. 1996. Boltzmann Gibbs weights in the branching random walk.In Classical and Modern Branching Processes K. B. Athreya and P. Jagers, eds. 84 41 50. Springer, New York. Z. Z. · Zbl 0866.60074
[9] COHN, H. 1985. A martingale approach to supercritical CMJ branching processes. Ann. Probab. 13 1179 1191. · Zbl 0587.60086
[10] DURRETT, R. and LIGGETT, M. 1983. Fixed points of the smoothing transform.Wahrsch. Verw. Gebiete 64 275 301. Z. · Zbl 0506.60097
[11] FELLER, W. 1971. An Introduction to Probability Theory and Its Applications, 2nd ed. Wiley, New York. · Zbl 0219.60003
[12] HEYDE, C. C. 1970. Extension of a result of Seneta for the supercritical branching process. Ann. Math. Statist. 41 739 742. · Zbl 0195.19201
[13] JAGERS, P. 1975. Branching Processes with Biological Applications. Wiley, New York. · Zbl 0356.60039
[14] JAGERS, P. 1989. General branching processes as Markov fields. Stochastic Process. Appl. 32 183 212. · Zbl 0678.92009
[15] KAHANE, J. P. and PEYRIERE, J. 1976. Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22 131 145. · Zbl 0349.60051
[16] KINGMAN, J. F. C. 1975. The first birth problem for an age-dependent branching process. Ann. Probab. 3 790 801. · Zbl 0325.60079
[17] KURTZ, T. G. 1972. Inequalities for the law of large numbers. Ann. Math. Statist. 43 1874 1883. · Zbl 0251.60019
[18] LIU, Q. 1996. Fixed points of a generalized smoothing transformation and applications to branching processes. Unpublished manuscript.
[19] LIU, Q. 1997. Sur une equation fonctionelle et ses applications: une extension du theoreme de Kesten Stigum concernant des processus de branchement. Adv. in Appl. Probab. 29. JSTOR: · Zbl 0901.60055
[20] LYONS, R. 1996. A simple path to Biggins’ martingale convergence. In Classical and ModernBranching Processes K. B. Athreya and P. Jagers, eds. 84 217 222. Springer, New York. Z. Z. · Zbl 0897.60086
[21] NERMAN, O. 1981. On the convergence of supercritical general C-M-J branching process.Wahrsch. Verw. Gebiete 57 365 395. Z. · Zbl 0451.60078
[22] NEVEU, J. 1988. Multiplicative martingales for spatial branching processes. In Seminar onStochastic Processes, 1987 E. C \?inlar, K. L. Chung and R. K. Getoor, eds. Prog. Probab. Statist. 15 223 241. Birkhauser, Boston. \" Z. · Zbl 0652.60089
[23] PAKES, A. G. 1992. On characterizations via mixed sums. Austral. J. Statist. 34 323 339. · Zbl 0759.62006
[24] SENETA, E. 1968. On recent theorems concerning the supercritical Galton Watson process. Ann. Math. Statist. 39 2098 2102. · Zbl 0176.47603
[25] WAYMIRE, E. C. and WILLIAMS, S. C. 1994. A general decomposition theory for random cascades.Bull. Amer. Math. Soc. N. S. 31 216 222. Z. · Zbl 0805.60045
[26] WAYMIRE, E. C. and WILLIAMS, S. C. 1995. Multiplicative cascades: dimension spectra and dependence. J. Fourier Analysis and Applications. Special issue in honour of J.-P. Kahane 589 609. · Zbl 0889.60050
[27] WAYMIRE, E. C. and WILLIAMS, S. C. 1996. A cascade decomposition theory with applications to Markov and exchangeable cascades. Trans. Amer. Math. Soc. 384 585 632. JSTOR: · Zbl 0857.60028
[28] SHEFFIELD, S3 7RH UNITED KINGDOM E-MAIL: j.biggins@sheffield.ac.uk
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.