Limit theorems for the maximal tree size of a Galton – Watson forest in the critical case

Abstract

Abstract We consider a critical Galton – Watson branching process starting with N particles; the number of offsprings is supposed to have the distribution p k =(k + 1)−τ −(k + 2)−τ , k=0, 1, 2, … Limit distributions of the maximal tree size are obtained for the corresponding Galton – Watson forest with N trees and n non-root vertices as N, n → ∞, n/N τ ⩾ C > 0.