For a supercritical branching processes with immigration {Zn};it is known that under suitable conditions on the offspring and immigration distributions, Zn/mn converges almost surely to a finite and strictly positive ...For a supercritical branching processes with immigration {Zn};it is known that under suitable conditions on the offspring and immigration distributions, Zn/mn converges almost surely to a finite and strictly positive limit, where m is the offspring mean. We are interested in the limiting properties of P(Zn=kn) with kn=o(mn) as n→∞. We give asymptotic behavior of such lower deviation probabilities in both Schröder and Böttcher cases, unifying and extending the previous results for Galton-Watson processes in literature.展开更多
We consider non-extinct branching processes in general random environments. Under the condition of means and second moments of each generation being bounded, we give the upper bounds and lower bounds for some form dev...We consider non-extinct branching processes in general random environments. Under the condition of means and second moments of each generation being bounded, we give the upper bounds and lower bounds for some form deviations of the process.展开更多
基金This work was supported in part by the National Natural Science Foundation of China(Grant No.11871103)the National Key Research and Development Program of China(No.2020YFA0712900)Research Foundation for Youth Scholars of Beijing Technology and Business University(Grant No.PXM2019_014213_000007).
文摘For a supercritical branching processes with immigration {Zn};it is known that under suitable conditions on the offspring and immigration distributions, Zn/mn converges almost surely to a finite and strictly positive limit, where m is the offspring mean. We are interested in the limiting properties of P(Zn=kn) with kn=o(mn) as n→∞. We give asymptotic behavior of such lower deviation probabilities in both Schröder and Böttcher cases, unifying and extending the previous results for Galton-Watson processes in literature.
基金Supported by National Natural Science Foundation of China （Grant No. 11026088）, Zhejiang Provincial Natural Science Foundation of China （Grant No. Y6100663）, ZJPEDF （Grant No. Y200906909）
文摘We consider non-extinct branching processes in general random environments. Under the condition of means and second moments of each generation being bounded, we give the upper bounds and lower bounds for some form deviations of the process.
