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2019 Invariance principle for non-homogeneous random walks
Nicholas Georgiou, Aleksandar Mijatović, Andrew R. Wade
Electron. J. Probab. 24: 1-38 (2019). DOI: 10.1214/19-EJP302

Abstract

We prove an invariance principle for a class of zero-drift spatially non-homogeneous random walks in ${\mathbb R}^d$, which may be recurrent in any dimension. The limit $\mathcal{X} $ is an elliptic martingale diffusion, which may be point-recurrent at the origin for any $d\geq 2$. To characterize $\mathcal{X} $, we introduce a (non-Euclidean) Riemannian metric on the unit sphere in ${\mathbb R}^d$ and use it to express a related spherical diffusion as a Brownian motion with drift. This representation allows us to establish the skew-product decomposition of the excursions of $\mathcal{X} $ and thus develop the excursion theory of $\mathcal{X} $ without appealing to the strong Markov property. This leads to the uniqueness in law of the stochastic differential equation for $\mathcal{X} $ in ${\mathbb R}^d$, whose coefficients are discontinuous at the origin. Using the Riemannian metric we can also detect whether the angular component of the excursions of $\mathcal{X} $ is time-reversible. If so, the excursions of $\mathcal{X} $ in ${\mathbb R}^d$ generalize the classical Pitman–Yor splitting-at-the-maximum property of Bessel excursions.

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Nicholas Georgiou. Aleksandar Mijatović. Andrew R. Wade. "Invariance principle for non-homogeneous random walks." Electron. J. Probab. 24 1 - 38, 2019. https://doi.org/10.1214/19-EJP302

Information

Received: 23 April 2018; Accepted: 27 March 2019; Published: 2019
First available in Project Euclid: 18 May 2019

zbMATH: 07068779
MathSciNet: MR3954788
Digital Object Identifier: 10.1214/19-EJP302

Subjects:
Primary: 60J05, 60J60
Secondary: 58J65, 60F17, 60J55

JOURNAL ARTICLE
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Vol.24 • 2019
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