Transitional phenomena in critical branching processes in a random environment: critical and subcritical cases

Let $\{Z_n, n \in \mathbb{N}_0\}$ be a critical branching process in a random environment $\Xi$. We consider the perturbation of this process, given by triangular array scheme of branching processes $\{Z_{k,n}, k \leq n\}$ with the same random environment $\Xi$. Denote by $b_{k,n}$, $k\leq n$, the difference of the associated random walks of $Z_{k,n}$ and $Z_k$.
We show that if $b_{k,n} = o(\sqrt{k})$ as $k \to \infty$, then
\begin{equation} \label{eq1} \mathsf{P}\left(Z_{n,n} > 0\right) \sim \mathsf{P}\left(Z_n > 0\right), \; n \to \infty. \end{equation}

However, if $b_{k,n} = - g(k / n) \sqrt{n}$ for some non-negative function $g(x)$, $x \in [0, 1]$, and for all $k \leq n$, then
\begin{equation} \label{eq2} \mathsf{P}\left(Z_{n,n} > 0\right) \sim \gamma \mathsf{P}\left(Z_n > 0\right), \; n \to \infty, \end{equation}
where the constant $\gamma \in (0, 1)$ depends on $g(x)$, $x \in [0, 1]$.