The number Yn of offspring of the most prolific individual in the nth generation of a Bienaymé–Galton–Watson process is studied. The asymptotic behaviour of Yn as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems for both Yn and EYn provided that the offspring mean is finite are obtained using some convergence results for branching processes as well as a transfer limit lemma for maxima. Subcritical, critical and supercritical branching processes are considered separately.

The mathematical model we consider here is the classical Bienaymé–Galton–Watson branching process modified with immigration in the state zero.

We study properties of the waiting time to explosion of the supercritical modified process, i.e. that time until all beginning cycles which die out have disappeared. We then derive the expected total progeny of a cycle and show how higher moments can be computed. With a view to applications the main goal is to show that any statistical inference from observed cycle lengths or estimates of total progeny on the fertility rate of the process must be treated with care. As an example we discuss population experiments with trout.

This paper presents some limit theorems for the simple branching process allowing immigration, {Xn}, when the offspring mean is infinite. It is shown that there exists a function U such that {e–nU/(Xn)} converges almost surely, and if s = ∑ bj, log+U(j) < ∞, where {bj} is the immigration distribution, the limit is non-defective and non-degenerate but is infinite if s = ∞.

When s = ∞, limit theorems are found for {U(Xn)} which involve a slowly varying non-linear norming.