Abstract:
Let $\mathcal{Z} = (Z_n, n \geq 0)$ be a branching process in a random environment (BPRE).
In 2020, A.V. Shklyaev obtained the asymptotic behavior of large deviation probabilities of $\ln Z_n$ in integro-local and integral forms. Also he showed the connection with the associated random walk distribution.
Approximately at the same time, similar results were obtained by D. Buraczewski, P. Dyszewski and E.I. Prokopenko, M.A. Struleva using other methods.
We will consider branching processes in a random environment with cooling (BPREC) that were introduced by V.A. Vatutin and firstly described by I.D. Korshunov in 2023.
A BPREC differs from a usual BPRE in such way that each value of the environment is determined for several generations.
Let us consider a more general model
for obtaining the asymptotic behavior of the large deviation probabilities of BPREC.
Let $\{\zeta_i, i\in\mathbb{N}\}$ be independent identically distributed lattice random variables,
$\{\tau_i, i\in\mathbb{N}\}$ be a bounded deterministic positive integer sequence.
Let us define $k(n)$ for any fixed $n$ via the following relation
$$
k(n) = \max\{j: \tau_1 + \dotsb + \tau_{j-1} < n\}.
$$
A random recurrent sequence is such sequence $Y_n$ that satisfies the equation
\begin{gather*}
Y_n = A_n Y_{n-1} + B_n
\end{gather*}
where $A_{n} = \exp \left(\zeta_{k(n)}\right)$, $B_n$ is some sequence independent of the future (values of $A_{k}$ after
the next moment $\tau_1 + \dotsc + \tau_n$).
Note that $B_i$ can be dependent on each other.
In the case of $\tau_i = 1$, the model represents the model of a linear recurrent sequence introduced by A.V. Shklyaev.
Under some moment assumptions on $A_n$ and $B_n$ we obtained the large deviation probabilities
$\mathbf{P}(\ln{Y_n} \in [x, x + \Delta_n))$ where
$\Delta_n$ is a positive sequence tending to zero slowly enough as $n \to \infty$ and $x$ corresponds to the first zone of large deviations.
The described result allows us to get the asymptotic behavior of the large deviation probabilities of BPREC.
The representation of BPREC as a random recurrent sequence allows us to get similar results for BPREC.
Note that we consider the case of bounded $\tau_i$. The next step is to consider the case when $\tau_i$ increases with growth of $i$.
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