Athreya K B
Iowa State University, Ames 50011.
J Math Biol. 1992;30(6):577-81.
If qk is the extinction probability of a slightly supercritical branching process with offspring distribution [pkr:r = 0, 1, 2, ...], then it is shown that if supk sigma r r3pkr less than infinity, inf sigma 2k greater than 0, and mk----1, then 1-qk approximately 2(mk - 1)sigma -2k, where mk = sigma r rpkr, sigma 2k = k sigma r r2pkr - m2k. This provides a simple set of sufficient conditions for the validity of a conjecture of Ewens (1969) for the survival probability of a slightly advantageous mutant gene.
若(q_k)是具有后代分布({p_{kr}:r = 0, 1, 2, \ldots})的轻度超临界分支过程的灭绝概率，则可以证明，如果(\sup_k\sum_{r}r^3p_{kr} \lt \infty)，(\inf\sigma^2_k \gt 0)，且(m_k \to 1)，那么(1 - q_k \approx 2(m_k - 1)\sigma^{-2}k)，其中(m_k = \sum{r}rp_{kr})，(\sigma^2_k = \sum_{r}r^2p_{kr} - m^2_k)。这为埃文斯（1969年）关于轻度有利突变基因生存概率的一个猜想的有效性提供了一组简单的充分条件。
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