Abstract:
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 $g(x)$, $x \in [0, 1]$.
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