Passage-times for partially-homogeneous reflected random walks on the quadrant

TL;DR

This work analyzes partially homogeneous reflecting random walks in the quadrant with zero interior drift, establishing a sharp recurrence/transience classification and precise tail bounds for return (passage) times. The authors reduce the interior covariance to a canonical form via a linear transform $T_\Sigma$, converting the interior into a wedge of angle $\varphi_0=\arccos(-\rho)$ and introducing the boundary-reflection angles $\varphi_1,\varphi_2$ and the composite parameter $\chi=(\varphi_1+\varphi_2)/\varphi_0$, which governs the asymptotics. They prove stabilization of effective boundary drifts, construct harmonic Lyapunov functions on the transformed wedge, and use time-changed processes to derive upper and lower tails for $\tau_Z(r)$, with exponents determined by $\chi$ and the interior covariance. As applications, the results yield quantitative passage-time tails for multidimensional Lindley processes and their mirror-reflected analogues, extending prior recurrence classifications to quantitative tail behavior. The methods blend harmonic analysis on wedges, Doeblin ratio limits, and Foster–Lyapunov techniques to obtain sharp, dimensioned tail exponents and moment criteria.

Abstract

We consider a random walk on the first quadrant of the square lattice, whose increment law is, roughly speaking, homogeneous along a finite number of half-lines near each of the two boundaries, and hence essentially specified by finitely-many transition laws near each boundary, together with an interior transition law that applies at sufficient distance from both boundaries. Under mild assumptions, in the (most subtle) setting in which the mean drift in the interior is zero, we classify recurrence and transience and provide power-law bounds on tails of passage times; the classification depends on the interior covariance matrix, the (finitely many) drifts near the boundaries, and stationary distributions derived from two one-dimensional Markov chains associated to each of the two boundaries. As an application, we consider reflected random walks related to multidimensional variants of the Lindley process, for which the recurrence question was studied recently by Peigné and Woess (Ann. Appl. Probab., vol. 31, 2021) using different methods, but for which no previous quantitative results on passage-times appear to be known.

Figures (3)

• Figure 1: The quadrant (left) and its image under the transformation $T_\Sigma$ (right) as a wedge with angular span $\varphi_0$ defined at \ref{['eq:inangle']}. Also depicted are the inwards-pointing normal vectors to the boundaries (dashed lines), the effective boundary drifts (dotted lines, $\overline{\mu}_1, \overline{\mu}_2$), their transformed counterparts ($v_1, v_2$), and the angles of the latter relative to the appropriate normal vectors $(\varphi_1, \varphi_2)$; the sign conventions are such that in the case indicated in the figure, $\varphi_1 >0$ and $\varphi_2 < 0$. In this example $\sigma_1^2 =\sigma_2^2 =3$, $\kappa =-1$, $\rho = -1/3$, $\overline{\mu}_1 = (-1, 1)$, and $\overline{\mu}_2 = (2,1)$ (see §\ref{['sec:linear_transformation']} for definitions of $\sigma_1, \sigma_2, \rho$ in terms of entries in $\Sigma$). The ellipse on the left represents the correlation structure of $\Sigma$ (its shape is the locus of $x \mapsto \Sigma x$ over $x \in \mathbb{S}$); the circle on the right corresponds to the identity matrix. In the contrary case $\rho > 0$, the wedge angle $\varphi_0$ is obtuse.
• Figure 2: The relationship between parameters $\beta_1, \beta_2$ and the gradient $\nabla h$ at $z_1 = (1,0)$ and $z_2 = (\cos \varphi_0, \sin \varphi_0)$. The unit inwards-pointing normal vectors at the boundaries are $n_1 = (0,1)$ and $n_2 = (\sin \varphi_0, - \cos \varphi_0)$. The derivative formulas \ref{['eq:h-derivatives']} show that when $\theta =0$, $\nabla h$ is in the direction $(\cos \beta_1, \sin \beta_1)$, while when $\theta = \varphi_0$, $\nabla h$ is in the direction $(\cos (\varphi_0 - \beta_2), \sin (\varphi_0 - \beta_2))$.
• Figure 3: An illustration of partial homogeneity and the identification of the boundary transition function $p_1(y;z)$ for the increment started from $(x,y) \in \mathbb{X}_1$, for the mirror-reflected walk $M$ (left) and the Lindley walk $L$ (right). For $M$, the probability of an increment of size $(z_1,z_2)$ is the sum of the probability of the direct jump from the circle to the square plus the probability of the reflected jump from the circle to the triangle. For $L$, the probability to make an increment of $(z_1,-y)$ includes all increments $(z_1,w)$, with $w$ such that $w + y \leq 0$, the action of taking the positive part of $y+w$ represented by the dotted line.

Theorems & Definitions (42)

• Definition 1.1: Lindley random walk
• Definition 1.2: Mirror-reflected random walk
• Theorem 1.3
• Remark 1.4
• Remark 2.2
• Theorem 2.3
• Proposition 2.5
• Example 3.1: Orthogonal reflection
• Proposition 4.1: Stabilization
• Lemma 4.2