再论非齐次生灭过程有真分布的条件

THE RESTATEMENT OF THE CONDITIONS OF NON-HOMOGENEUOUS BIRTH AND DEATH PROCESSES WITH SURE DISTRIBUTION

本文给出了使生灭过程{X(t),t≥0}有真分布的充分条件与必要条件,所谓真分布即 sum i=0 to ∞ p_1(t)=1,其中 p_1(t)=P(X(t)=i),所获得的结果与[1]、[5]的主要区别在于所有条件都是加在生灭系数之和之上的,且特别强调了生灭微分方程解的唯一性的至关重要性,此外在较宽松的条件下彻底推广了Feller-Lundberg 定理。

In this paper,we give the necessary or sufficent conditions respectively such that a non-homogeneuous Birth and Death process {X(t),t>0} has sure distribution,that is,sum from i-0 to ∞ P_i(t)=1,where P,(t)=P(X(t)=i).The main difference from [1][6] is that all conditions are dependent on the sums of so called birth and death coefficients,and that we cmphasize the importance of the uniqueness of the solution of Birth and Death differential equation system corresponding to the process,in addition,we extend ultimately Feller-Lundberg theorem.