Let $Z(n)$ be a Bienaymé–Galton–Watson (BGW) system with offspring generating function (GF) $f(s) = {\mathbb E}[{s^X}]$ for $s \in [0,1]$ , mean $m = f'(1 - ) \in (0,1)$ (subcritical case), and iterates ${f_n} = {f^{ \circ n}}$ . Denote the survival probability

\begin{equation*}{Q_n}\,:\!=\, {\mathbb P}\{ Z(n) \gt 0\} = 1 - {f_n}(0),\end{equation*}

and assume $2b\,:\!=\, f''(1 - ) \lt \infty $ . The Kolmogorov constant is

\begin{equation*}{\mathcal{K}} = \mathop {\lim }\limits_{n \to \infty } \frac{{{Q_n}}}{{{m^n}}} \in (0,1]{\kern 1pt} .{\kern 1pt} \end{equation*}

Theorem 1.2 of [Reference Imomov and Murtazaev1] is stated for every such $f$ , and asserts that

(1) \begin{equation}{\mathcal{K}} = {{\mathcal{K}}_{\rm{I}}}\,:\!=\, \frac{{m(1 - m)}}{{m(1 - m) + b}}{\kern 1pt} ,{\kern 1pt} \end{equation}

written in [Reference Imomov and Murtazaev1] in the equivalent form

\begin{equation*}{\mathcal{K}} = \frac{1}{{1 + b/(m(1 - m))}}{\kern 1pt} .{\kern 1pt} \end{equation*}

The range of validity of that theorem is exactly the linear-fractional class. The equality (1) holds precisely when $f$ is of linear-fractional form, so that the formula of Theorem 1.2 is exact on that class and only on it; for subcritical GFs outside that class, ${\mathcal{K}}$ may be strictly larger or strictly smaller than ${{\mathcal{K}}_{\rm{I}}}$ . Beyond this, ${{\mathcal{K}}_{\rm{I}}}$ retains a quantitative meaning: within the class of embeddable offspring GFs one has

\begin{equation*}{\mathcal{K}} \ge {{\mathcal{K}}_{\rm{I}}},\end{equation*}

with equality if and only if $f$ is linear-fractional, as follows from Theorem 3.1 of [Reference Pakes2]. Thus the local data $m$ and $b$ determine a sharp bound for ${\mathcal{K}}$ , but do not determine ${\mathcal{K}}$ itself. A detailed discussion is given in [Reference Pakes2].

Accordingly, Theorem 1.2 of [Reference Imomov and Murtazaev1], together with the noncritical statement derived from it there, is to be read under the additional hypothesis that the offspring GF is linear-fractional, and every assertion of [Reference Imomov and Murtazaev1] depending on that theorem is to be restricted correspondingly.