[Paper Review] On recurrence of reflected random walk on the half-line. With an appendix on results of Martin Benda

Marc Peigné, Wolfgang Woess|ArXiv.org|Dec 12, 2006

Stochastic processes and statistical mechanics4 references7 citations

TL;DR

This paper establishes recurrence criteria for reflected random walks on the half-line by analyzing the embedded process of reflections. It introduces an invariant measure for this embedded process and proves that recurrence occurs if and only if the quadratic tail condition $\int_0^\infty \mathbb{P}[Y_1 \geq t]^2 dt < \infty$ holds, which generalizes previous results and applies even when $\mathbb{E}[Y_1] = \infty$. The key contribution is a sharp moment condition for recurrence in both non-negative and signed increment cases.

ABSTRACT

Let $(Y_n)$ be a sequence of i.i.d. real valued random variables. Reflected random walk $(X_n)$ is defined recursively by $X_0=x \ge 0$, $X_{n+1} = |X_n - Y_{n+1}|$. In this note, we study recurrence of this process, extending a previous criterion. This is obtained by determining an invariant measure of the embedded process of reflections.

Motivation & Objective

• To extend the recurrence criterion for reflected random walks on the half-line beyond the classical case where $\mathbb{E}[Y_1] < \infty$.
• To analyze the embedded process of reflections—reflected random walk observed only at reflection times—as a Markov chain with an invariant measure.
• To determine necessary and sufficient conditions for recurrence based on the tail behavior of the increment distribution.
• To generalize results to cases where increments may be negative, including centered increments.
• To provide a rigorous foundation using unpublished results from Martin Benda on stochastic iterated function systems and local contractivity.

Proposed method

• Define the reflected random walk via $X_{n+1} = |X_n - Y_{n+1}|$ with $X_0 \geq 0$ and i.i.d. increments $Y_n$.
• Introduce the process of reflections $R_k = X_{\mathbf{r}(k)}$, where $\mathbf{r}(k)$ is the time of the $k$-th reflection, forming a Markov chain on $[0, \infty)$.
• Derive the transition kernel of the reflection process: $q(x, B) = \int_{[0,x)} \mu(B + x - w) \mathcal{U}(dw)$ for $x > 0$, with $\mathcal{U} = \sum_{n=0}^\infty \mu^{(n)}$.
• Establish that the existence of a finite invariant measure for the reflection process is equivalent to the quadratic tail condition $\int_0^\infty \mathbb{P}[Y_1 \geq t]^2 dt < \infty$.
• Use results from Martin Benda on stochastic iterated function systems (SFS) and local contractivity to analyze pathwise convergence of coupled processes.
• Prove that if the semigroup generated by the contractions $g_y(x) = |x - y|$ contains a constant function, then the SFS is locally contractive, implying pathwise convergence of coupled trajectories.

Experimental results

Research questions

• RQ1 Under what conditions is a reflected random walk on the half-line recurrent when $\mathbb{E}[Y_1] = \infty$?
• RQ2 How does the quadratic tail condition $\int_0^\infty \mathbb{P}[Y_1 \geq t]^2 dt < \infty$ relate to the existence of a finite invariant measure for the embedded reflection process?
• RQ3 What is the recurrence behavior of reflected random walks when increments may be negative, particularly in the centered case?
• RQ4 How can the theory of stochastic iterated function systems and local contractivity be used to analyze recurrence in reflected processes?
• RQ5 What is the role of the invariant measure of the reflection process in determining recurrence of the original reflected walk?

Key findings

• The reflected random walk is recurrent if and only if the quadratic tail condition $\int_0^\infty \mathbb{P}[Y_1 \geq t]^2 dt < \infty$ holds, which is equivalent to the existence of a finite invariant measure for the embedded reflection process.
• When $Y_n \geq 0$ a.s., recurrence holds if $\mathbb{E}[\sqrt{Y_1}] < \infty$, which implies the quadratic tail condition.
• For centered increments ($\mathbb{E}[Y_1^-] = \mathbb{E}[Y_1^+]$), recurrence holds under the condition $\mathbb{E}\bigl[\sqrt{Y_1^+}^3\bigr] < \infty$, which is shown to be almost sharp.
• The invariant measure for the reflection process is constructed via the potential measure $\mathcal{U} = \sum_{n=0}^\infty \mu^{(n)}$, and its finiteness determines recurrence.
• The proof relies on the local contractivity of the stochastic iterated function system generated by the maps $x \mapsto |x - y|$ and the fact that the constant function $0$ lies in the semigroup generated by these maps.
• The recurrence of the original process follows from the recurrence of the embedded reflection process, which is established via the existence of a finite invariant measure and pathwise convergence of coupled trajectories.