带移民的上临界分枝过程的下偏差研究

On Lower Deviation Probabilities of Supercritical Branching Processes with Immigration

本文考虑带移民的上临界Galton-Watson过程{X_(n)}_(n)>0.众所周知,存在常数列{cn}n≥0,使得当n→∞时,X_(n)/c_(n)几乎处处收敛到某个随机变量V.本文利用Cramér变换,给出了过程{X_(n)}_(n)≥0的下偏差,即概率P(X_(n)=k)当k充分大时的渐近性质,其中k_(n)≤k≤C_(n)且k_(n)→∞.

This paper considers a supercritical Galton-Watson process with immigra-tion{X_(n)}n≥0.It is well-known that there is a sequence of constants{C_(n)}n≥0 such that X_(n)/c_(n)→V almost surely as n→oo.Using Cramer transforms,we obtain lower devi-ations for the process(X_(n))_(n)≥0o,which refer to the asymptotic properties of P(X_(n)=k)for sufficiently large k satisfying kn≤k≤C_(n) and k_(n)→∞.