Long-time behaviour of supercritical finite circular mechanism branching processes

Abstract Let {Zna*b:n≥ 0} be a discrete-time branching process with circular mechanism a*b. For mechanism a, the offspring distribution is {aj:j≥ 0}. For mechanism b, the offspring distribution is {bj:j ≥0}. Let ma=∑j≥0 jaj and mb=∑j≥0 jbj . The extinction property of such branching processes is first studied. It is proved that Wn=Zna*b/γn ( γn=(mamb)k for n=2k and γn=(mamb)kma for n=2k+1) is an integrable martingale and hence converges to some random variable W. Then, under assumption that a0=b0=0, a1,b1>0 and aj,bj≠1 for any j≠ 1, we study the rates of convergence to zero as k→∞ of P(|Z2k+1a*b/Z2ka*b-ma |>ε), P(|Z2ka*b/Z2k-1a*b-mb |>ε), P(|Wk-W|>ε) P(|Z2k+1a*b/Z2ka*b-ma |>ε|W>δ), P(|(Z2ka*b/Z2k-1a*b-mb |>ε|W>δ) for ε>0 and δ>0 under various moment conditions on {aj} and {bj}. It is shown that the rates for the first two are geometric while the last three rates are always supergeometric under a finite moment generating function hypothesis.