Conditional Functional Limit Theorem for a Random Recurrence Sequence Conditioned on a Large Deviation Event

Theory of Probability &Its Applications, Volume 69, Issue 1, Page 99-116, May 2024.
Let $\{Z_n,\, n\ge 0\}$ be a branching process in an independent and identically distributed (i.i.d.) random environment and $\{S_n,\, n\,{\ge}\, 1\}$ be the associated random walk with steps $\xi_i$. Under the Cramér condition on $\xi_1$ and moment assumptions on a number of descendants of one particle, we know the asymptotics of the large deviation probabilities $\mathbf{P}(\ln Z_n > x)$, where $x/n > \mu^*$. Here, $\mu^*$ is a parameter depending on the process type. We study the asymptotic behavior of the process trajectory under the condition of a large deviation event. In particular, we obtain a conditional functional limit theorem for the trajectory of $(Z_{[nt]},\, t\in [0,1])$ given $\ln Z_n>x$. This result is obtained in a more general model of linear recurrence sequence $Y_{n+1}=A_n Y_n + B_n$, $n\ge 0$, where $\{A_i\}$ is a sequence of i.i.d. random variables, $Y_0$, $B_i$, $i\ge 0$, are possibly dependent and have different distributions, and we need only some moment assumptions on them.