论文标题
迭代的扰动随机步行的稳定波动在一般分支过程的中间世代中
Stable fluctuations of iterated perturbed random walks in intermediate generations of a general branching process tree
论文作者
论文摘要
考虑一个通用的分支过程，又称crump-mode-jagers进程，由扰动的随机步行$η_1$，$ξ_1+η_2$，$ξ_1+ξ_2+η_3，\ ldots $生成。在这里，$（ξ_1，η_1）$，$（ξ_2，η_2），\ ldots $是具有任意依赖性正组件的独立分布的随机向量。用$ n_j（t）$表示$ j $ j $ th Generation time $ \ leq t $的人数。假设$ j = j（t）\ to \ infty $和$ j（t）= o（t^a）$ as $ t \ to \ to \ to \ infty $对于某些明确给定的$ a> 0 $（在论文中指定）。相应的$ j $ th一代属于中级世代集。我们提供了足够的条件，在该条件下，该过程的有限维分布$（n _ {\ lfloor j（t）u \ rfloor}（t））_ {u> 0} $，适当地归一化和中心化，弱收敛到具有有限lévyvy的稳定LévyVyVyvy均值的整体功能。
Consider a general branching process, a.k.a. Crump-Mode-Jagers process, generated by a perturbed random walk $η_1$, $ξ_1+η_2$, $ξ_1+ξ_2+η_3,\ldots$. Here, $(ξ_1,η_1)$, $(ξ_2, η_2),\ldots$ are independent identically distributed random vectors with arbitrarily dependent positive components. Denote by $N_j(t)$ the number of the $j$th generation individuals with birth times $\leq t$. Assume that $j=j(t)\to\infty$ and $j(t)=o(t^a)$ as $t\to\infty$ for some explicitly given $a>0$ (to be specified in the paper). The corresponding $j$th generation belongs to the set of intermediate generations. We provide sufficient conditions under which finite-dimensional distributions of the process $(N_{\lfloor j(t)u\rfloor}(t))_{u>0}$, properly normalized and centered, converge weakly to those of an integral functional of a stable Lévy process with finite mean.
