Large and moderate deviations for a class of renewal random indexed branching process

Let {Zn,n=0,1,2,…}{Zn,n=0,1,2,…} be a supercritical branching process, {Nt,t≥0}{Nt,t≥0} be a renewal process independent of {Zn,n=0,1,2,…}{Zn,n=0,1,2,…}, then {ZNt,t≥0}{ZNt,t≥0} is a supercritical renewal random indexed branching process. Assume that {Nt,t≥0}{Nt,t≥0} has a Erlang(2) renewal distribution, we show large and moderate deviation principles for logZNtlogZNt.