On decoupled standard random walks

Let Sₙ=∑ₖ₌₁ⁿξₖ, n, be a standard random walk with i.i.d. nonnegative increments ξ₁,ξ₂,… and associated renewal counting process N(t)=∑n≥ 11_ₙ≤ t\, t≥ 0. A decoupling of (Sₙ)n≥ 1 is any sequence Ŝ₁, Ŝ₂,… of independent random variables such that, for each n, Ŝₙ and Sₙ have the same law. Under the assumption that the law of Ŝ₁ belongs to the domain of attraction of a stable law with finite mean, we prove a functional limit theorem for the decoupled renewal counting process N̂(t)=∑n≥ 11_̂ₙ≤ t\, t≥ 0, after proper scaling, centering and normalization. We also study the asymptotics of log P≥ 1Ŝₙ>t\ as t→∞ under varying assumptions on the law of Ŝ₁. In particular, we recover the assertions which were previously known in the case when Ŝ₁ has an exponential law. These results, which were formulated in terms of an infinite Ginibre point process, served as an initial motivation for the present work. Finally, we prove strong law of large numbers type results for the sequence of decoupled maxima Mₙ=max1≤ k≤ nŜₖ, n, and the related first passage time process τ(t)=inf: Mₙ>t\, t≥ 0. In particular, we provide a tail condition on the law of Ŝ₁ in the case when the latter has finite mean but infinite variance that implies limt→∞t⁻¹τ(t)=limt→∞t⁻¹Eτ(t)=0. In other words, t⁻¹τ(t) may exhibit a different limit behavior than t⁻¹τ(t), where τ(t) denotes the level-t first passage time of (Sₙ)n≥ 1.