Geodesic
Conditional functional limit theorem for random reccurence sequence conditioned on large deviation event
Teoriâ veroâtnostej i ee primeneniâ, Tome 69 (2024) no. 1, pp. 125-147 Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: large deviations, functional limit theorem, branching processes, bisexual branching processes, random environment. 
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     author = {A. V. Shklyaev},
     title = {Conditional functional limit theorem for random reccurence sequence conditioned on large deviation event},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {125--147},
     year = {2024},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2024_69_1_a6/}
}
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A. V. Shklyaev. Conditional functional limit theorem for random reccurence sequence conditioned on large deviation event. Teoriâ veroâtnostej i ee primeneniâ, Tome 69 (2024) no. 1, pp. 125-147. http://geodesic.mathdoc.fr/item/TVP_2024_69_1_a6/

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