Weak convergence for the minimal position in a branching random walk: A simple proof
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AK CONVERGENCE FOR THE MINIMAL POSITION IN A BRANCHING RANDOM WALK: A SIMPLE PROOF ´kon1 and Zhan Shi2 Elie A¨ide 1
Department of Mathematics and Computer Science, Technische Universiteit Eindhoven P.O. Box 513, 5600 MB Eindhoven, The Netherlands E-mail: [email protected] 2
Laboratoire de Probabilit´es UMR 7599, Universit´e Paris VI 4 place Jussieu, F-75252 Paris Cedex 05, France E-mail: [email protected] (Received March 15, 2010; Accepted June 10, 2010)
Dedicated to Professors Endre Cs´ aki and P´ al R´ev´esz on the occasion of their 75th birthdays
Summary Consider the boundary case in a one-dimensional super-critical branching random walk. It is known that upon the survival of the system, the minimal position after n steps behaves in probability like 32 log n when n → ∞. We give a simple and self-contained proof of this result, based exclusively on elementary properties of sums of i.i.d. real-valued random variables.
1. Introduction Consider a (discrete-time, one-dimensional) branching random walk. It starts with an initial ancestor particle located at the origin. At time 1, the particle dies, producing a certain number of new particles; these new particles are positioned according to the law of a given finite point process. At time 2, these particles die, each giving birth to new particles that are positioned (with respect to the birth place) according to the law of the same point process. And the system goes on indefinitely, as long as there are particles that are alive. We assume that each particle produces new particles independently of other particles in the same generation, and of everything up to that generation. Mathematics subject classification number : 60J80. Key words and phrases: branching random walk, minimal position. 0031-5303/2010/$20.00
c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
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´ E. A¨IDEKON and Z. SHI
The number of particles in each generation obviously forms a Galton–Watson process, which will always be assumed to be super-critical. Let (V (x), |x| = n) be the positions of the particles at the n-th generation. The process (V (x)) indexed by a Galton–Watson tree is called a branching random walk. We do not assume the random variables V (x), |x| = 1, to be independent, nor necessarily identically distributed, though it is often assumed in the literature (for example, in [16]). We are interested in min|x|=n V (x), the minimal position of the branching random walk after n steps. Under a mild integrability assumption, we have (Kingman [11], Hammersley [7], Biggins [2]), on the set of non-extinction, 1 min V (x) → γ, n |x|=n
a.s.,
(1.1)
where γ ∈ R is a known constant. The rate of convergence in (1.1) has recently been studied, independently, by Hu and Shi [8], and Addario-Berry and Reed [1]. To state the result, we assume the following condition: E E (1.2) e−V (x) = 1, V (x)e−V (x) = 0. |x|=1
|x|=1
This is referred to in the literature as the boundary case; see for example Biggins and Kyprianou [3]. Under (1.2), we have γ = 0 in
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