On decoupled standard random walks

Let S_n=∑_k=1^nξ_k, n∈ℕ, be a standard random walk with i.i.d. nonnegative increments ξ_1,ξ_2,… and associated renewal counting process N(t)=∑_n≥ 11_{S_n≤ t}, t≥ 0. A decoupling of (S_n)_n≥ 1 is any sequence Ŝ_1, Ŝ_2,… of independent random variables such that, for each n∈ℕ, Ŝ_n and S_n have the same law. Under the assumption that the law of Ŝ_1 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_{Ŝ_n≤ t}, t≥ 0, after proper scaling, centering and normalization. We also study the asymptotics of logℙ{min_n≥ 1Ŝ_n>t} as t→∞ under varying assumptions on the law of Ŝ_1. In particular, we recover the assertions which were previously known in the case when Ŝ_1 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 M_n=max_1≤ k≤ nŜ_k, n∈ℕ, and the related first passage time process τ̂(t)=inf{n∈ℕ: M_n>t}, t≥ 0. In particular, we provide a tail condition on the law of Ŝ_1 in the case when the latter has finite mean but infinite variance that implies lim_t→∞t^-1τ̂(t)=lim_t→∞t^-1𝔼τ̂(t)=0. In other words, t^-1τ̂(t) may exhibit a different limit behavior than t^-1τ(t), where τ(t) denotes the level-t first passage time of (S_n)_n≥ 1.