Limit Theorems in Bellman–Harris Processes with Finites Second Moments

Abstract

In this article are studied different theorems limits in a critical Bellman-Harris branching process with a only type of particle and with finite second moments. There were used two processes in order to figure out the limits as following as: “The condition of no extinction” and “The condition of extinction in the near future”. In the two previous processes is taken into account two different cases as: τ_i := d_it y τ_i := d_i ± t, where t is a point of time and di ∈ (0,∞) are fixed for every i = 1, . . . , k. For the case where τ_i := d_it, the Esty’s comparison lemma 2.3 is used to investigate the asymptotic behavior of the joint probability generating function F (s₁, . . . , τ_k), for t → ∞; for the case τ_i := t + d_i, is not used. For this last case is founded another comparison lemma (lemma 4.3), that is the base to demonstrate the theorems limits if τ_i := t ± d_i.