Stable fluctuations of iterated perturbed random walks in intermediate generations of a general branching process tree

TL;DR

This paper studies the fluctuations of intermediate generations in a general branching process driven by a perturbed random walk, showing convergence to a stable Lévy process under certain conditions.

Contribution

It provides new conditions for the weak convergence of normalized intermediate generation counts to a stable Lévy process.

Findings

Finite-dimensional distributions converge to an integral of a stable Lévy process.

Results apply to a broad class of perturbed random walks with dependent components.

Provides a framework for understanding fluctuations in complex branching structures.

Abstract

Consider a general branching process, a.k.a. Crump-Mode-Jagers process, generated by a perturbed random walk $\eta_1$, $\xi_1+\eta_2$, $\xi_1+\xi_2+\eta_3,\ldots$. Here, $(\xi_1,\eta_1)$, $(\xi_2, \eta_2),\ldots$ 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 L\'{e}vy process with finite mean.