Approximation of Fluctuations in a Sequence of Nearly Critical Branching Processes

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.

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