摘要
Let{Zn,n≥0}be a supercritical branching process in an independent and identically distributed random environment.We prove Cramer moderate deviations and Berry-Esseen bounds for log(Zn+n0/Zn0)uniformly in n0∈N,which extend the corresponding results by I.Grama,Q.Liu,and M.Miqueu[Stochastic Process.Appl.,2017,127:1255-1281]established for n0=0.The extension is interesting in theory,and is motivated by applications.A new method is developed for the proofs;some conditions of Grama et al.are relaxed in our present setting.An example of application is given in constructing confidence intervals to estimate the criticality parameter in terms of log(Zn+n0/Zn0)and n.
基金
supported by the National Natural Science Foundation of China(Grant Nos.11601375,11971063,11731012)
the Natural ScienceFoundation of Guangdong Province(Grant No.2018A030313954)
the Centre Henri Lebesgue(CHL,ANR-11-LABX-0020-01).
