Limit theorems for globally perturbed random walks

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.