An application of the coalescence theory to branching random walks

Abstract

In a discrete-time single-type Galton--Watson branching random walk {Z_n, ζ_n}_{n≤ 0}, where Z_n is the population of the nth generation and ζ_n is a collection of the positions on ℝ of the Z_n individuals in the nth generation, let Y_n be the position of a randomly chosen individual from the nth generation and Z_n(x) be the number of points in ζ_n that are less than or equal to x for x∈ℝ. In this paper we show in the explosive case (i.e. m=E(Z₁∣ Z₀=1)=∞) when the offspring distribution is in the domain of attraction of a stable law of order α,0 <α<1, that the sequence of random functions {Z_n(x)/Z_n:−∞<x<∞} converges in the finite-dimensional sense to {δ_x:−∞<x<∞}, where δ_x≡ 1_{{N≤ x}} and N is an N(0,1) random variable.