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Fluctuations of Discrete-Time Random Walks

Denis Denisov, Vitali Wachtel

TL;DR

This work analyzes fluctuations of one-dimensional random walks in the oscillating regime, focusing on first-passage times and their conditional distributions. It contrasts classical Wiener–Hopf factorisation with a robust universality approach that leverages functional limit theorems and martingale methods to transfer Brownian-exit-time insights to pre-limit walks, including time-inhomogeneous increments and Markov chains. The results yield precise tail asymptotics for exit times, conditional limit theorems, and local limit theorems for conditioned walks, and they establish harmonic functions and Doob $h$-transforms for walks killed at leaving the positive half-line. The framework extends to i.i.d. and non-i.i.d. settings, providing a versatile toolkit for boundary-crossing problems and contributing a unified probabilistic viewpoint beyond generating-function techniques.

Abstract

These notes are devoted to fluctuations of one-dimensional random walks. We discuss various approaches to first-passage times and to the corresponding conditional distributions. After discussion of some classical methods, such as reflection principle for simple random walks and Wiener-Hopf factorisation, we proceed to the universality approach, which has been developed in recent past. Considering one-dimensional case allows us to avoid some technical obstacles and to present the core of this method in a more transparent way. It turns out that the universality method is much more robust than the Wiener-Hopf factorisation and allows one to consider walks with non-identically distributed or even dependent increments.

Fluctuations of Discrete-Time Random Walks

TL;DR

This work analyzes fluctuations of one-dimensional random walks in the oscillating regime, focusing on first-passage times and their conditional distributions. It contrasts classical Wiener–Hopf factorisation with a robust universality approach that leverages functional limit theorems and martingale methods to transfer Brownian-exit-time insights to pre-limit walks, including time-inhomogeneous increments and Markov chains. The results yield precise tail asymptotics for exit times, conditional limit theorems, and local limit theorems for conditioned walks, and they establish harmonic functions and Doob -transforms for walks killed at leaving the positive half-line. The framework extends to i.i.d. and non-i.i.d. settings, providing a versatile toolkit for boundary-crossing problems and contributing a unified probabilistic viewpoint beyond generating-function techniques.

Abstract

These notes are devoted to fluctuations of one-dimensional random walks. We discuss various approaches to first-passage times and to the corresponding conditional distributions. After discussion of some classical methods, such as reflection principle for simple random walks and Wiener-Hopf factorisation, we proceed to the universality approach, which has been developed in recent past. Considering one-dimensional case allows us to avoid some technical obstacles and to present the core of this method in a more transparent way. It turns out that the universality method is much more robust than the Wiener-Hopf factorisation and allows one to consider walks with non-identically distributed or even dependent increments.
Paper Structure (18 sections, 46 theorems, 479 equations)

This paper contains 18 sections, 46 theorems, 479 equations.

Table of Contents

  1. Introduction
  2. Simple Random Walks
  3. Left-continuous random walks.
  4. Dual stopping times and Wiener--Hopf factorisation
  5. A Universality Approach to Exit Times
  6. Upper bounds for the tail of $T_g$.
  7. Some martingale identities.
  8. Estimates in the boundary problem.
  9. Proof of Theorem \ref{['th:3.1']}.
  10. Conditional functional limit theorem and some properties of the function $U_g$
  11. Conditioned random walks with i.i.d. increments.
  12. Local Limit Theorem for Conditional Distributions
  13. Markov chains
  14. Construction of the positive harmonic function
  15. Coupling of Markov chain and a simple random walk
  16. ...and 3 more sections

Key Result

Proposition 2.1

Let $\{S(n)\}$ be a symmetric simple random walk. Then, for all $x,y\ge1$, and

Theorems & Definitions (90)

  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • proof
  • Corollary 2.5
  • proof
  • ...and 80 more