There are determined the asymptotic behavior of the survival probability of a critical branching process in a random environment. In the special case of independent identically distributed geometric offspring distributions, and the somewhat more general case of offspring distributions with linear fractional generating functions, M. V. Kozlov
[Theory Probab. Appl. 21(1976), 791-804 (1977); translation from Teor. Veroyatn. Primen. 21, 813-825 (1976; Zbl 0384.60058
)] proved that, as
, the probability of nonextinction at generation
is proportional to
. There are established Kozlov’s asymptotic for general independent identically distributed offspring distributions.