Limit theorem for a supercritical Galton-Watson process

Abstract

Letμ n, n = 0, 1, ..., be a Galton-Watson process, and τx + 1 the instant of first crossing of the level x by the process. A limit theorem is proved for the joint distribution of the random variables $$\tau _x ,x - \mu _{\tau _x } ,\mu _{\tau _x + 1} - x(x \to \infty )$$ on the assumption that Mμ1 In (1 + μ1) < ∞.