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(Z 1∣ Z 0=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.