Abstract:
Let （
Z0,
Z1,
Z2, …） be a Galton-Watson process and
T denote its extinction time, i. e.
T=
K \Leftrightarrow
Zk-1＞1
Zk=0,
bj==\lim _n \rightarrow \infty P\left(Z_n=j \mid n
m＜1, EZ1 logZ1=+∞, then \overline\lim _n \rightarrow \infty \fraca_nm^n\left(1-f_n(s)\right) \leqslant 1-\mathscrB(s), （2） If \lim _n \rightarrow \infty \sum_j=1^k \fracb_ja_n \cdot \frac1-\left(1-m^n\right)^jm^n<+\infty, then 1-fn（s）=0（mn/an）, （3）\lim _n \rightarrow \infty \fracq-f_n(s)r^n=\fracq\left(1-\mathscrB\left(\fracsq\right)\right)\sum_j=1^\infty j b_j, for m≠1 and EZ1 log Z1＜+∞, （4） We also prove two useful results by （3）, i. e. Theorem 3 and Theorem 4.
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