全状态上临界Athreya－Karlin　B．P．R．E．的灭种概率的渐近行为

The Asymptotic Behaviour of Extinction Probability in the Total-state Supereritical Athreya-Karlin B.P.R.E.

当随机环境为一有限状态平稳遍历马氏链时，本文证明了总存在常数Ｃ１、Ｃ２、＞０，使得全状态上临界Ａｔｈｒｅｙａ－ＫａｒｌｉｎＢ．Ｐ．Ｒ．Ｅ．当初始个体数ｋ增大趋于无穷时相应的灭种概率ｑｋ的渐近变化总介于和之间，其中０＜β＜１；α１≥α２≥０都是由所给过程完全确定的常数。

Abstract It is shown that when the random environment is a stationary ergodic Markov chain with a finite state space,as the initial population size k tends to infinity,there exist constants ci,c2>0 such that the asymptotic behaviour of the corresponding extinction probability qk of a total-state supercritical Athreya-Karlin B.P.R.E.is always bounded between c1βkK-α1 and C2βkKα2,where 0 <β<1;α1≥α2≥0 are constants determined by the B.P.R.E..