First passage times for decoupled random walks

TL;DR

This paper studies the asymptotic behavior of first passage times and maxima of decoupled random walks, revealing five regimes with distinct limit theorems and connecting to the Ginibre point process.

Contribution

It establishes distributional convergence of first passage times and maxima of decoupled random walks, identifying five different regimes and their limit processes.

Findings

Existence of five regimes with distinct limit theorems.

First passage times converge to inverse extremal-like processes.

Number of visits converges to stationary Gaussian processes.

Abstract

Motivated by a connection to the infinite Ginibre point process, decoupled random walks were introduced in a recent article Alsmeyer, Iksanov and Kabluchko (2025). The decoupled random walk is a sequence of independent random variables, in which the $n$th variable has the same distribution as the position at time $n$ of a standard random walk with nonnegative increments. We prove distributional convergence in the Skorokhod space equipped with the $J_1$-topology of the running maxima and the first passage times of decoupled random walks. We show that there exist five different regimes, in which distinct limit theorems arise. Rather different functional limit theorems for the number of visits of decoupled standard random walk to the interval $[0,t]$ as $t\to\infty$ were earlier obtained in the aforementioned paper Alsmeyer, Iksanov and Kabluchko (2025). While the limit processes for the first passage times are inverse extremal-like processes, the limit processes for the number of visits are stationary Gaussian.