Resumen de An application of the coalescence theory to branching random walks.

K.B. Athreya, J. Y. Y.-I. Hong

• In a discrete-time single-type Galton--Watson branching random walk {Zn, ?n}n=0, where Zn is the population of the nth generation and ?n is a collection of the positions on of the Zn individuals in the nth generation, let Yn be the position of a randomly chosen individual from the nth generation and Zn (x) be the number of points in ?n that are less than or equal to x for x ? R. In this paper we show in the explosive case (i.e. m = ? (Z1 | Z0 = 1) = 8 when the offspring distribution is in the domain of attraction of a stable law of order or, 0 < a < 1, that the sequence of random functions {Zn(x)/Zn : - 8 < x < 8} converges in the finite-dimensional sense to {dx: - 8 < x < 8}, where dx = 1{N=x} and N is an N(0, 1) random variable.