On the time of attaining a maximum by a critical branching process in a random environment and by a stopped random walk

Let $\{\xi_n\}$ be a critical branching process in a random environment with linear-fractional generating functions, $T$ be the time of extinction of $\{\xi_n\}$, $T_M$ be the first maximum passage time of $\{\xi_n\}$. We study the asymptotic behaviour of $\mathsf P(T_M>n)$ and prove limit theorems for the random variables $\{T_M/n\mid T>n\}$ and $\{T_M/T\mid T>n\}$ as $n\to\infty$. Similar results are established for the stopped random walk with zero drift.