Electronic Journal of Probability

Invariance principle for non-homogeneous random walks

Nicholas Georgiou, Aleksandar Mijatović, and Andrew R. Wade

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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.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 48, 38 pp.

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

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Primary: 60J05: Discrete-time Markov processes on general state spaces 60J60: Diffusion processes [See also 58J65]
Secondary: 60F17: Functional limit theorems; invariance principles 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 60J55: Local time and additive functionals

non-homogeneous random walk invariance principle diffusion limits excursions skew product rapid spinning recurrence transience

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Georgiou, Nicholas; Mijatović, Aleksandar; Wade, Andrew R. Invariance principle for non-homogeneous random walks. Electron. J. Probab. 24 (2019), paper no. 48, 38 pp. doi:10.1214/19-EJP302. https://projecteuclid.org/euclid.ejp/1558145016

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