Long-Time Behavior of Galton–Watson Systems with Circular Mechanism

Let {Zn:n≥0} be a Galton–Watson system with a circular mechanism a∗b, where a={aj}j=0∞ and b={bj}j=0∞ are probability distributions on Z+:={0,1,2,⋯}. Let ma:=∑j=0∞jaj, mb:=∑j=0∞jbj. The extinction property of such branching systems is first studied. Then, it is proved that there exists Γn such that Wn=Γn−1Zn is an integrable martingale and hence converges to some random variable W. Moreover, for the case that a0=b0=0, a1,b1∈(0,1), the convergence rates to 0 of PZn+1Zn−ωn>ε∣Z0=1,P(|Wn−W|>ε∣Z0=1) and PZn+1Zn−ωn>ε∣W≥δ,Z0=1 as n→∞ are presented for ε,δ>0 under various moment conditions on {aj} and {bj}, where ωn=ma or mb for n being even or odd, respectively. It is further shown that the first rate is geometric while the last two rates are supergeometric under a finite moment generating function hypothesis.