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+αL(1 − s) where α ∈ (0, 1) and L(x) varies slowly as x → 0+. Then we find, as t → ∞, (P{Z(t)> 0})αL(P{Z(t)>0})∼ μ/αt where μ 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 + uα)−1/α for u ⩾/ 0. Moment conditions on the lifetime distribution required for the above results are discussed.