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First passage times for decoupled random walks

Alexander Iksanov, Zakhar Kabluchko, Vitali Wachtel

TL;DR

This work analyzes decoupled random walks, where each $\hat S_n$ shares the same distribution as the $n$-th position of a random walk with nonnegative increments, and derives five regimes for the tail of the increment distribution that yield distinct functional limit behaviors for the running maxima and first passage times. By proving weak convergence in the Skorokhod $J_1$ topology via vague convergence to Poisson random measures, the authors obtain explicit limit processes $X_1,X_2,X_3,X_4$ and their inverse-time variants, with detailed one-dimensional marginals. The results reveal a stark contrast with classical RW: the decoupled setting produces inverse extremal-like limits for first passage times and stationary Gaussian limits for visit-counts in related settings, with the maxima leading to Fréchet-type laws. The analysis also connects to renewal theory and motivates the numerical structure through the infinite Ginibre point process, enriching the understanding of how tail behavior governs extreme and time-to-extinction phenomena in decoupled systems.

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.

First passage times for decoupled random walks

TL;DR

This work analyzes decoupled random walks, where each shares the same distribution as the -th position of a random walk with nonnegative increments, and derives five regimes for the tail of the increment distribution that yield distinct functional limit behaviors for the running maxima and first passage times. By proving weak convergence in the Skorokhod topology via vague convergence to Poisson random measures, the authors obtain explicit limit processes and their inverse-time variants, with detailed one-dimensional marginals. The results reveal a stark contrast with classical RW: the decoupled setting produces inverse extremal-like limits for first passage times and stationary Gaussian limits for visit-counts in related settings, with the maxima leading to Fréchet-type laws. The analysis also connects to renewal theory and motivates the numerical structure through the infinite Ginibre point process, enriching the understanding of how tail behavior governs extreme and time-to-extinction phenomena in decoupled systems.

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 th variable has the same distribution as the position at time of a standard random walk with nonnegative increments. We prove distributional convergence in the Skorokhod space equipped with the -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 as 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.
Paper Structure (12 sections, 14 theorems, 113 equations, 2 figures)

This paper contains 12 sections, 14 theorems, 113 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. The first passage time into $(t,\infty)$ vs the number of visits to $(-\infty, t]$
  3. Proofs of the main results
  4. Large deviation probabilities in the case $\alpha\le 2$
  5. Proof of Theorem \ref{['thm:02']}
  6. Proof of Theorem \ref{['thm:03']}
  7. Proof of Theorem \ref{['thm:05']}
  8. Proof of Theorem \ref{['thm:06']}
  9. Proof of Theorem \ref{['thm:02tau']}
  10. Proofs of Theorems \ref{['thm:03tau']} and \ref{['thm:06tau']}
  11. Proof of Theorem \ref{['thm:05tau']}
  12. Appendix

Key Result

Theorem 1.1

Assume that for some $\alpha\in (0,2]$ and some $\ell$ slowly varying at $\infty$. Let $a(v)$ be any positive function satisfying $\lim_{v\to\infty}v^2\mathbb{P}\{\xi>a(v)\}=1$. Denote by $(t_k, j_k)$ the atoms of a Poisson random measure $\mathcal{P}_\alpha^{(1)}$ on $[0,\infty)\times (0,\infty]$ with mean mea If $\alpha\in (0,2)$ or $\alpha=2$ and $\lim_{v\to\infty}\ell(v)=\infty$, then in the

Figures (2)

  • Figure 1: Left: A sample path of the process $(X_1(t))_{t\ge 0}$ from Theorem \ref{['thm:02']}, together with the points $(t_k,j_k)$ of the underlying Poisson random measure $\mathcal{P}_\alpha^{(1)}$. The vertical axis is displayed on a logarithmic scale. Right: A sample path of the process $(X_2(t))_{t\in\mathbb{R}}$ from Theorem \ref{['thm:03']}, together with the points $(t_k,\mu t_k+j_k)$.
  • Figure 2: Left: A sample path of the process $(X_3(t))_{t\in \mathbb{R}}$ from Theorem \ref{['thm:05']}, together with the atoms $(t_k,j_k)$ of the underlying Poisson random measure $\mathcal{P}^{(3)}$. Right: A sample path of the process $(X_4(t))_{t\in\mathbb{R}}$ from Theorem \ref{['thm:06']}, together with the points $(t_k,\mu t_k+j_k)$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Lemma 3.1
  • ...and 14 more