AN APPLICATION OF THE COALESCENCE THEORY TO BRANCHING RANDOM WALKS

In a discrete-time single-type Galton Watson branching random walk {Z(n), zeta(n)}(n >= 0), where Z(n) is the population of the nth generation and zeta(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 zeta(n) that are less than or equal to x for x is an element of R. In this paper we show in the explosive case (i.e. m = E(Z(1)vertical bar Z(0) = 1) = infinity) when the offspring distribution is in the domain of attraction of a stable law of order alpha, 0 < alpha < 1, that the sequence of random functions {Z(n)(x)/Z(n) : -infinity < x < infinity} converges in the finite-dimensional sense to {delta(x) : -infinity < x < infinity}, where delta(x) 1({N <= x}) and N is an N(0,1) random variable.