Remarks on Kolmogorov’s constant for simple branching processes

Abstract Recent investigations have argued that there is a simple explicit representation for the Kolmogorov constant c associated with the subcritical Galton–Watson branching process. We exhibit examples showing that although this representation can be valid, it more often is not. Our work is presented in terms of the limiting conditional mean population size $\mu=c^{-1}$ . The analogous quantity for the Markov branching process is denoted by $\widehat\mu$ . We show that the simple representation put forward for $\widehat\mu$ in fact is an upper bound that is attained only if the offspring-number probability-generating function is quadratic. The conditional mean $\mu$ is the limit of a computable increasing sequence $(\mu_n$ ). Estimates of n are determined ensuring that, for any small positive number $\varepsilon$ , $0\lt\mu-\mu_n\le \varepsilon$ .