论文标题
迭代扰动随机步行在一般分支过程树上的续订理论：中级世代
Renewal theory for iterated perturbed random walks on a general branching process tree: intermediate generations
论文作者
论文摘要
令$（ξ_k，η_k）_ {k \ in \ mathbb {n}} $是独立分布的随机向量，随机向量具有任意依赖的正组件。我们称（全球）扰动随机步行a随机序列$（t_k）_ {k \ in \ mathbb {n}} $由$ t_k：=ξ_1+\ cdots+cdots+C+ξ_{k-1}+η_k$ for $ k \ in \ MathBb in \ Mathbb {n n n} $。此外，通过迭代的扰动随机步行是指确定一个普通分支过程中个体的出生时间的点过程的序列，前提是第一代人的出生时间是通过扰动的随机行走给出的。对于$ j \ in \ mathbb {n} $和$ t \ geq 0 $，用$ n_j（t）$表示$ j $ j $ th Generation Systy具有出生时间$ \ leq t $的人数。在本文中，我们证明了$ n_j（t）$的经典更新理论结果（基本续签定理，Blackwell定理和关键续订定理），假设$ j = j（t）\ to \ j（t）\ to \ infty $ j = j（t）\ to \ infty $ and $ j（t）= \ j（t）= o（t）= o（t）= o（t）= o（t）= o（t）根据我们的术语，这些世代构成了中间世代集的子集。
Let $(ξ_k,η_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $(T_k)_{k\in\mathbb{N}}$ defined by $T_k:=ξ_1+\cdots+ξ_{k-1}+η_k$ for $k\in\mathbb{N}$. 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 $j\in\mathbb{N}$ and $t\geq 0$, 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, Blackwell's theorem and the key renewal theorem) for $N_j(t)$ under the assumption that $j=j(t)\to\infty$ and $j(t)=o(t^{2/3})$ as $t\to\infty$. According to our terminology, such generations form a subset of the set of intermediate generations.
