Limit theorems for globally perturbed random walks

TL;DR

This paper investigates the asymptotic behavior of a class of perturbed random walks, establishing laws of large numbers and functional limit theorems under various moment conditions, with applications to passage times and counting processes.

Contribution

It provides new limit theorems for perturbed random walks, including weak and strong laws, and characterizes the conditions for convergence to Brownian motion.

Findings

Weak law of large numbers for passage time $ au(t)$

Functional limit theorems for $( au(ut))$, $(N(ut))$, and $( ho(ut))$

Conditions for convergence to Brownian motion

Abstract

Let $(\xi_1, \eta_1)$, $(\xi_2, \eta_2),\ldots$ be independent copies of an $\mathbb{R}^2$-valued random vector $(\xi, \eta)$ with arbitrarily dependent components. Put $T_n:= \xi_1+\ldots+\xi_{n-1} + \eta_n $ for $n\in\mathbb{N}$ and define $\tau(t) := \inf\{n\geq 1: T_n>t\}$ 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 $\rho(t):=\sup\{n\geq 1: T_n \leq t\}$ the associated last exit time for $t\in\mathbb{R}$. The standing assumption of the paper is $\mathbb{E}[\xi]\in (0,\infty)$. We prove a weak law of large numbers for $\tau(t)$ and strong laws of large numbers for $\tau(t)$, $N(t)$ and $\rho(t)$. The strong law of large numbers for $\tau(t)$ holds if, and only if, $\mathbb{E}[\eta^+]<\infty$. In the complementary situation $\mathbb{E}[\eta^+]=\infty$ we prove functional limit theorems in the Skorokhod space for $(\tau(ut))_{u\geq 0}$, properly normalized without centering. Also, we provide sufficient conditions under which finite dimensional distributions of $(\tau(ut))_{u\geq 0}$, $(N(ut))_{u\geq 0}$ and $(\rho(ut))_{u\geq 0}$, 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 $ut/\mathbb{E}[\xi]$ needed for $(\tau(ut))$ and $(\rho(ut))$. Finally, we prove a functional limit theorem in the Skorokhod space for $(N(ut))$ under optimal moment conditions.