The asymptotic behaviour of extinction probability in the Smith.Wilkinson branching process

Under some regularity conditions, in the supercritical Smith.Wilkinson branching process it is shown that as k, the starting population size, tends to infinity, the rate of convergence of qk, the corresponding extinction probability, to zero is similar to that of: k.., if there exists at least one subcritical state in the random environment space; xkk.., if there exist only supercritical states in the random environment space; exp , if there exists at least one critical state and the others are supercritical in the random environment space. Here ., x, . and c are positive constants determined by the process.