Laws of the iterated logarithm for iterated perturbed random walks

TL;DR

This work establishes laws of the iterated logarithm for both a globally perturbed random walk and its iterated (branched) counterparts. It removes the previously needed finite moments assumption on the perturbations $\eta$ and proves LILs for real-time counts $Y(t)$ as $t\to\infty$, as well as for the generation-wise counts $Y_j(t)$ in iterated perturbed random walks with mean functions $V_j(t)$. The authors develop a decomposition into renewal-type components and negligible remainder terms, employ exponential-martingale techniques, smoothing of distribution functions, and Brownian-approximation arguments for renewal fluctuations, and leverage regular variation of convolution structures. The results significantly extend prior work by handling arbitrary $\eta$ distributions and by establishing ultimate LIL behavior for all generations in the iterated setting, with precise normalization constants depending on $\sigma^2=\mathrm{Var}(\xi)$ and $\mu=\mathbb{E}[\xi]$. This advances understanding of extreme fluctuations in perturbed and iterated stochastic processes with broad applicability in branching-type models and occupancy schemes.

Abstract

Let $(ξ_k, η_k)_{k\geq 1}$be independent identically distributed random vectors with arbitrarily dependent positive components and $T_k:=ξ_1+\ldots+ξ_{k-1}+η_k$for $k\in\mathbb{N}$. We call the random sequence {T_k, k=1,2...} a (globally) perturbed random walk. Consider a general branching process generated by {T_k, k=1,2...} and let Y_j(t) denote the number of the jth generation individuals with birth times less or equal t. Assuming that Var ξ_1 is finite and allowing the distribution of η_1 to be arbitrary, we prove a law of the iterated logarithm (LIL) for Y_j(t). In particular, a LIL for the counting process of {T_k, k=1,2...} is obtained. The latter result was previously established in the article Iksanov, Jedidi and Bouzeffour (2017) under the additional assumption that Eη^a is finite for some positive a. In this paper, we show that the aforementioned additional assumption is not needed.

Theorems & Definitions (8)

• Theorem 1
• Theorem 2
• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Proposition 1