THE TOTAL NUMBER OF PARTICLES IN A REDUCED BELLMAN-HARRIS BRANCHING-PROCESS

Let z(t) be the number of particles at time t in a Bellman-Harris branching process with generating function f (s) of the number of direct descendants and distribution G(t) of particle lifelength satisfying the conditions f'(1) = 1, f(s) = s + (1 - s) 1+alpha L(1 - S), where alpha is-an-element-of (0, 1], the function L(x) varies slowly as x --> 0+, and lim(n-->infinity) n(1 - G(n))/1 - f(n)(0) = 0, where f(n)(s) is the nth iteration of f(s). Denote by {z(tau, t), 0 less-than-or-equal-to tau less-than-or-equal-to t} the corresponding reduced Bellman-Harris branching process, where z(tau, t) is the number of particles in the initial process at time tau whose descendants or they themselves are alive at time t. Let nu(t) be the number of dead particles of the reduced process to time t. The paper finds the limiting distribution of nu(t) under the conditions z(t) > 0 and t --> infinity.