Limit theorems for globally perturbed random walks

TL;DR

This work analyzes a globally perturbed random walk with positive drift, examining the first passage time $\tau(t)$, the number of visits $N(t)$, and the last exit time $\rho(t)$. It establishes weak and strong laws for these functionals and, under optimal moment and tail conditions, functional limit theorems in Skorokhod spaces, including Brownian-type limits and heavy-tailed inverse-extremal limits. A key finding is that $N(t)$ may require a two-term centering to attain a Brownian limit, unlike $\tau(t)$ and $\rho(t)$ which center at $t/\mathbb{E}[ξ]$, highlighting distinct limit regimes driven by the perturbations $\eta$. The results combine maximal-process limit theorems, inversion of first-passage functionals, and martingale/shot-noise techniques, yielding a comprehensive picture of the asymptotic behavior of globally perturbed random walks.

Abstract

Let $(ξ_1, η_1)$, $(ξ_2, η_2),\ldots$ be independent copies of an $\mathbb{R}^2$-valued random vector $(ξ, η)$ with arbitrarily dependent components. Put $T_n:= ξ_1+\ldots+ξ_{n-1} + η_n $ for $n\in\mathbb{N}$ and define $τ(t) := \inf\{n\geq 1: T_n>t\}$ the first passage time into $(t,\infty)$, $N(t) :=\sum_{n\geq 1}1_{\{T_n\leq t\}}$ the number of visits to $(-\infty, t]$ and $ρ(t):=\sup\{n\geq 1: T_n \leq t\}$ the associated last exit time for $t\in\mathbb{R}$. The standing assumption of the paper is $\mathbb{E}[ξ]\in (0,\infty)$. We prove a weak law of large numbers for $τ(t)$ and strong laws of large numbers for $τ(t)$, $N(t)$ and $ρ(t)$. The strong law of large numbers for $τ(t)$ holds if, and only if, $\mathbb{E}[η^+]<\infty$. In the complementary situation $\mathbb{E}[η^+]=\infty$ we prove functional limit theorems in the Skorokhod space for $(τ(ut))_{u\geq 0}$, properly normalized without centering. Also, we provide sufficient conditions under which finite dimensional distributions of $(τ(ut))_{u\geq 0}$, $(N(ut))_{u\geq 0}$ and $(ρ(ut))_{u\geq 0}$, properly normalized and centered, converge weakly as $t\to\infty$ to those of a Brownian motion. Quite unexpectedly, the centering needed for $(N(ut))$ takes in general a more complicated form than the centering $ut/\mathbb{E}[ξ]$ needed for $(τ(ut))$ and $(ρ(ut))$. Finally, we prove a functional limit theorem in the Skorokhod space for $(N(ut))$ under optimal moment conditions.

Theorems & Definitions (25)

• Proposition 1.1
• Lemma 1.2
• Theorem 2.1
• Theorem 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Theorem 2.6
• Proposition 2.7
• Theorem 2.8