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: $i := dit$ y $i := di ± \tau $, where t is a point of time and $d_i\varepsilon (0,\infty )$ are fixed for every $i = 1, . . . ,k$. For the case where $i := di\tau$, the Esty’s comparison lemma 2.3 is used to investigate the asymptotic behavior of the joint probability generating function F$F(s_1,...,\tau_k)$, for $t\longrightarrow \infty $; for the case $i := \tau + di$ 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 $i := \tau ± di$.