Limit theorems for the critical age-dependent branching process with infinite variance

Let Z(t) be the population at time t of a critical age-dependent branching process. Suppose that the offspring distribution has a generating function of the form f(s) = s + (1 - s)1+[alpha]L(1 - s) where [alpha] [set membership, variant] (0, 1) and L(x) varies slowly as x --> 0+. Then we find, as t --> [infinity], (P{Z(t)> 0})[alpha]L(P{Z(t)>0})~ [mu]/[alpha]t where [mu] is the mean lifetime of each particle. Furthermore, if we condition the process on non-extinction at time t, the random variable P{Z(t)>0}Z(t) converges in law to a random variable with Laplace-Stieltjes transform 1 - u(1 + u[alpha])-1/[alpha] for u [greater-or-equal, slanted]/ 0. Moment conditions on the lifetime distribution required for the above results are discussed.