Critical branching processes in a varying environment and their asymptotiс behavior

A branching process in a varying environment (BPVE) extends the notion of a Galton-Watson process (GWP) in that the offspring distribution may vary among the generations. MacPhee and Schuh (1983) observed that a BPVE may exhibit kind of exotic behavior unknown for GWPs — like multiple rates of growth. In the talk we set forth that such peculiarities can be avoided by a mild second moment condition [1] which in succession allows a classification of BPVEs in the spirit of GWPs. In particular it turns out that — in contrast to critical branching processes in a random environment — critical BPVEs behave much the same as critical GWPs. For one thing this concerns the fundamental asymptotics due to Kolmogorov (1938) and Yaglom (1947) on the survival probabilities and the behavior of the conditioned processes, respectively. In the talk we rather focus on the genealogical structure and explain [2] that the well-known results on reduced critical GWPs by Zubkov (1975) and Fleischmann, Siegmund-Schultze (1977) carry over to critical BRVEs.