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The central limit theorem for the supercritical branching random walk, and related results. (English) Zbl 0703.60083

The standard branching random walk is a Galton-Watson process with one- dimensional spatial structure whereby the offspring of a parent take up positions relative to the parent according to a given point process, and different families behave independently. If \(Z^{(n)}\) denotes the n th generation point process (starting with a single ancestor at the origin) and \(N^{(n)}=Z^{(n)}({\mathbb{R}})\) is the size of the n th generation, then on the event \(\{N^{(n)}>0\}\), \(Z^{(n)}/N^{(n)}\) is a random probability measure on \({\mathbb{R}}\). The fact that the intensity measure of \(Z^{(n)}\) is an n-fold convolution suggests that this random probability measure behaves like that of the sum of n independent, identically distributed random variables, notably in respect of central limit or, more generally, stable properties.
This paper generalizes and strengthens earlier partial results in this area, such as those of N. Kaplan and S. Asmussen [Stochastic Processes Appl. 4, 15-31 (1976; Zbl 0322.60065)] and C. F. Klebaner [Adv. Appl. Probab. 14, 359-367 (1982; Zbl 0482.60078)].
Reviewer: D.R.Grey

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60F05 Central limit and other weak theorems
60E07 Infinitely divisible distributions; stable distributions
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[1] Asmussen, S.; Hering, H., Branching Processes (1983), Birkhäuser: Birkhäuser Boston · Zbl 0516.60095
[2] Athreya, K. D.; Kaplan, N., The additive property and its applications in branching processes, (Joffe, A.; Ney, P., Branching Processes (1978), Dekker: Dekker New York), 27-60 · Zbl 0404.60088
[3] Feller, W., An Introduction to Probability Theory and its Applications, Vol. II (1971), Wiley: Wiley New York · Zbl 0219.60003
[4] Gnedenko, B. V.; Kolmogorov, A. N., Limit Distributions for Sums of Independent Random Variables (1954), Addison-Wesley: Addison-Wesley Reading, MA, [English translation.] · Zbl 0056.36001
[5] Harris, T. E., The Theory of Branching Processes (1963), Springer: Springer Berlin · Zbl 0117.13002
[6] Kallenberg, O., Random Measures (1975), Akademie-Verlag: Akademie-Verlag Berlin, GDR · Zbl 0345.60031
[7] Kaplan, N.; Asmussen, S., Branching random walks II, Stochastic Process. Appl., 4, 15-31 (1976) · Zbl 0322.60065
[8] Klebaner, C. F., Branching random walk in varying environments, Adv. Appl. Probab., 14, 359-367 (1982) · Zbl 0482.60078
[9] Rvačeva, E. L., On domains of attraction of multidimensional distributions, Select. Trans. Math. Statist. Probab., 2, 183-205 (1962) · Zbl 0208.44401
[10] Stam, A. J., On a conjecture of Harris, Z. Wahrsch. Verw. Gebiete, 5, 202-206 (1966) · Zbl 0161.14503
[11] Stone, C., A local limit theorem for nonlattice multidimensional distribution functions, Ann. Math. Statist, 36, 546-551 (1965) · Zbl 0135.19204
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