THE BIRTH AND DEATH PROCESSES WITH ZERO AS THEIR ABSORBING BARRIER

Let E=(0,1,…),Q~b=(q_(ij)),i,j=0,1,…,where q_(i,i-1)=a_i,q_(i,i+1)=b_i,q_(ii)=-(a_i+b_i),q_(ij)=0,when|i-j|>1.a_0=0,b_0=b>0,a_i,b_i>0(i>0).Letting b=0 in Q~b,we get the matrix Q~0.The time homogeneous Markov process X~b={x~b(t,ω),0≤t<σ~b(ω)}(X~0={x~0(t,ω),0≤t<σ~0(ω)}),with Q~b(Q~0,respectively)as its density matrix and with E as its state space,is called Q~b(Q~0,respectively)process in this paper.Q~b and Q~0 processes are all called the birth and death processes,with zero being the reflecting barrier of Q~b processes,the absorbing barrier of Q~0 processes.All the Q~b processes have been constructed by both probability and analytical methods(Wang 2],Yang1]).In this paper,the Q~0 processes are“imbedded.”into Q~b processes and all the Q~0 processesare directly constructed from the Q~b processes.The main results are:Let b>0 be arbitrarily fixed,then there is a one to one correspondence between the Q~0 processesand the Q~b processes(in the sense of transition probability).The Q~0 process is uni