Limit theorems for globally perturbed random walks

Abstract

Let $(\hat{I}{¾}_1,\hat{I}{\cdot }_1)$, $(\hat{I}{¾}_2,\hat{I}{\cdot }_2),\dots$ be independent copies of an ${\mathbb{R}}^2$-valued random vector $(\hat{I}¾,\hat{I}\cdot )$ with arbitrarily dependent components. Put $T_n:=\hat{I}{¾}_1+\dots +\hat{I}{¾}_{n-1}+\hat{I}{\cdot }_n$ for $n\in \mathbb{N}$ and define $\ddot{I}„(t):=\inf \{n\ge 1:T_n>t\}$ the first passage time into $(t,\infty )$, $N(t):={\sum }_{n\ge 1}{1}_{\{T_n\le t\}}$ the number of visits to $(-\infty ,t]$ and $\ddot{I}(t):=\sup \{n\ge 1:T_n\le t\}$ the associated last exit time for $t\in \mathbb{R}$. The standing assumption of the paper is $\mathbb{E}[\hat{I}¾]\in (0,\infty )$. We prove a weak law of large numbers for $\ddot{I}„(t)$ and strong laws of large numbers for $\ddot{I}„(t)$, $N(t)$ and $\ddot{I}(t)$. The strong law of large numbers for $\ddot{I}„(t)$ holds if, and only if, $\mathbb{E}[\hat{I}{\cdot }^{+}]<\infty$. In the complementary situation $\mathbb{E}[\hat{I}{\cdot }^{+}]=\infty$ we prove functional limit theorems in the Skorokhod space for $(\ddot{I}„(ut){)}_{u\ge 0}$, properly normalized without centering. Also, we provide sufficient conditions under which finite dimensional distributions of $(\ddot{I}„(ut){)}_{u\ge 0}$, $(N(ut){)}_{u\ge 0}$ and $(\ddot{I}(ut){)}_{u\ge 0}$, 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 $ut/\mathbb{E}[\hat{I}¾]$ needed for $(\ddot{I}„(ut))$ and $(\ddot{I}(ut))$. Finally, we prove a functional limit theorem in the Skorokhod space for $(N(ut))$ under optimal moment conditions.

23 pages, submitted for publication to Stochastic Models