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

Abstract

Consider a general branching process, a.k.a. Crump-Mode-Jagers process, generated by a perturbed random walk ${\eta }_1$, ${\xi }_1+{\eta }_2$, ${\xi }_1+{\xi }_2+{\eta }_3,\dots$. Here, $({\xi }_1,{\eta }_1)$, $({\xi }_2,{\eta }_2),\dots$ 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 $\le t$. 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.

17 pages