Limit theorems for globally perturbed random walks

Let (<i>ξ</i>₁, <i>η</i>₁), (ξ2,η2),… be independent copies of an <i>ℝ</i>²-valued random vector (<i>ξ</i>, <i>η</i>) with arbitrarily dependent components. Put Tn:=ξ1+…+ξn−1+ηn for <i>n</i>∈<i>ℕ</i> and define τ(t):=inf{n≥1:Tn&gt;t} the first passage time into (<i>t</i>, ∞), N(t):=∑n≥1\mathds1{Tn≤t} the number of visits to (−∞,t] and ρ(t)∼:=∼sup{n≥1:Tn≤t} the associated last exit time for <i>t</i>∈<i>ℝ</i>. The standing assumption of the article is E[ξ]∈(0,∞). We prove a weak law of large numbers for <i>τ</i>(<i>t</i>) and strong laws of large numbers for <i>τ</i>(<i>t</i>), <i>N</i>(<i>t</i>), and <i>ρ</i>(<i>t</i>). The strong law of large numbers for <i>τ</i>(<i>t</i>) holds if, and only if, E[η+]&lt;∞. In the complementary situation E[η+]=∞, we prove functional limit theorems in the Skorokhod space for (<i>τ</i>(<i>u</i><i>t</i>))_<i>u</i>≥0, properly normalized without centering. Also, we provide sufficient conditions under which finite dimensional distributions of (<i>τ</i>(<i>u</i><i>t</i>))_<i>u</i>≥0, (<i>N</i>(<i>u</i><i>t</i>))_<i>u</i>≥0, and (<i>ρ</i>(<i>u</i><i>t</i>))_<i>u</i>≥0, properly normalized and centered, converge weakly as <i>t</i>→∞ to those of a Brownian motion. Quite unexpectedly, the centering needed for (<i>N</i>(<i>u</i><i>t</i>)) takes in general a more complicated form than the centering <i>u</i><i>t</i>∕<i>𝔼</i>[<i>ξ</i>] needed for (<i>τ</i>(<i>u</i><i>t</i>)) and (<i>ρ</i>(<i>u</i><i>t</i>)). Finally, we prove a functional limit theorem in the Skorokhod space for (<i>N</i>(<i>u</i><i>t</i>)) under optimal moment conditions.