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Asymptotics of returns to the coordinate hyperplanes for conditioned simple random walks

Rodolphe Garbit, Kilian Raschel

Abstract

In this paper we study the number of returns to the coordinate hyperplanes for multidimensional nearest-neighbour random walks. While one-dimensional results on returns are classical, much less is known in higher dimensions. We analyse the asymptotic behaviour of returns under several natural conditionings: the unconditioned walk, bridges, meanders, and non-negative bridges (or excursions). Our main results characterize the limiting distributions under appropriate rescaling. The resulting one-dimensional marginals may be half-normal, Rayleigh, geometric, negative binomial, or certain mixtures thereof. In most situations, the coordinates are asymptotically independent; however, there are notable exceptions for the meander case, depending on the drift. The proofs rely on conditioning on the numbers of horizontal and vertical steps, which restores a form of independence and reduces the problem to one-dimensional estimates via binomial convolution and Bernstein-type approximations.

Asymptotics of returns to the coordinate hyperplanes for conditioned simple random walks

Abstract

In this paper we study the number of returns to the coordinate hyperplanes for multidimensional nearest-neighbour random walks. While one-dimensional results on returns are classical, much less is known in higher dimensions. We analyse the asymptotic behaviour of returns under several natural conditionings: the unconditioned walk, bridges, meanders, and non-negative bridges (or excursions). Our main results characterize the limiting distributions under appropriate rescaling. The resulting one-dimensional marginals may be half-normal, Rayleigh, geometric, negative binomial, or certain mixtures thereof. In most situations, the coordinates are asymptotically independent; however, there are notable exceptions for the meander case, depending on the drift. The proofs rely on conditioning on the numbers of horizontal and vertical steps, which restores a form of independence and reduces the problem to one-dimensional estimates via binomial convolution and Bernstein-type approximations.
Paper Structure (25 sections, 17 theorems, 186 equations, 3 figures)

This paper contains 25 sections, 17 theorems, 186 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Context
  3. Main results
  4. Ingredients
  5. Binomial convolution
  6. Path decomposition and independence
  7. The method: An example
  8. Binomial convolution asymptotics
  9. Estimates and limit theorems for the simple random walk on Z
  10. Statements of one-dimensional limit theorems
  11. One-dimensional estimates
  12. Proofs of one-dimensional limit theorems
  13. Proof of Theorem \ref{['thm:convergencetohalfnormal']}
  14. Proof of Theorem \ref{['thm:bridgeconvergencecase']}
  15. Proof of Theorem \ref{['thm:excursionconvergencecase']}
  16. ...and 10 more sections

Key Result

Theorem 1

Set $a^i_n=\sqrt{h_i n}$ if $p_i=q_i$ and $a^i_n=1$ if $p_i\not= q_i$. As $n\to\infty$, it holds that where $X_1$ and $X_2$ are independent and the distribution of $X_i$ is

Figures (3)

  • Figure 1: Left: number of returns to $0$ (in red) for the simple random walk on $\mathbb{Z}$ with steps in $\{-1,0,1\}$ (often called a Motzkin walk in combinatorics). Right: number of returns for the same walk, now conditioned to stay non-negative. Here, the term "return" is used in a strict sense: the walk must leave the axis before returning to it. The number of returns is closely related to the total number of contacts with the axis, where every visit is counted.
  • Figure 2: Left: two ordered simple random walks on $\mathbb{N}_0$, exhibiting two types of boundary contacts: coincidences between the upper and lower paths, and contacts with the horizontal axis. Right: A simple random walk in the quarter plane; first returns to the coordinate axes are indicated in red.
  • Figure 3: Left: representation of the random walk $(S_n^1,S_n^2)_{n\geqslant 0}$. Middle: graph of $({\widetilde{S}}_n^1)_{n\geqslant 0}$. Right: graph of $({\widetilde{S}}_n^2)_{n\geqslant 0}$

Theorems & Definitions (28)

  • Theorem 1: Unconditioned case
  • Theorem 2: Bridge case
  • Theorem 3: Meander case
  • Theorem 4: Non-negative bridge
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Theorem 7: Unconditioned case
  • Theorem 8: Bridge case
  • ...and 18 more