In what follows all references to monographs are applicable also to multiauthorship volumes such as seminar notes.

We consider a sequence of discrete time branching processes with generation-dependent immigration, where the offspring mean tends to its critical value 1. Using a martingale approach, we prove functional limit theorems for suitable normalized fluctuations of the process around its mean when the mean number of immigrating individuals tends to infinity. The limiting processes are deterministically time-changed Wiener processes with three different non-linear time change functions, depending on the behavior of the mean and the variance of the number of immigrants. For the normalized sequence of processes we obtain a deterministic approximation. Consequences related to the maxima and the total progeny of the process will be discussed.

We consider a critical discrete-time branching process with generation dependent immigration. For the case in which the mean number of immigrating individuals tends to ∞ with the generation number, we prove functional limit theorems for centered and normalized processes. The limiting processes are deterministically time-changed Wiener, with three different covariance functions depending on the behavior of the mean and variance of the number of immigrants. As an application, we prove that the conditional least-squares estimator of the offspring mean is asymptotically normal, which demonstrates an alternative case of normality of the estimator for the process with nondegenerate offspring distribution. The norming factor is where α(n) denotes the mean number of immigrating individuals in the nth generation.

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.

Branching process, Time-dependent immigration, Skorokhod space, Offspring mean, Weighted estimator, Consistency, 60J80, 62F12, 60G99,