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تسجيل مستخدم جديدRenewal theory for iterated perturbed random walks on a general branching process tree: intermediate generations
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Let $(xi_k,eta_k)_{kinmathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $(T_k)_{kinmathbb{N}}$ defined by $T_k:=xi_1+cdots+xi_{k-1}+eta_k$ for $kinmathbb{N}$. Further, by an iterated perturbed random walk is meant the sequence of point processes defining the birth times of individuals in subsequent generations of a general branching process provided that the birth times of the first generation individuals are given by a perturbed random walk. For $jinmathbb{N}$ and $tgeq 0$, denote by $N_j(t)$ the number of the $j$th generation individuals with birth times $leq t$. In this article we prove counterparts of the classical renewal-theoretic results (the elementary renewal theorem, Blackwells theorem and the key renewal theorem) for $N_j(t)$ under the assumption that $j=j(t)toinfty$ and $j(t)=o(t^{2/3})$ as $ttoinfty$. According to our terminology, such generations form a subset of the set of intermediate generations.
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