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Invariance principle for non-homogeneous random walks

Georgiou, Nicholas; Mijatović, Aleksandar; Wade, Andrew R.

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Authors

Aleksandar Mijatović



Abstract

We prove an invariance principle for a class of zero-drift spatially non-homogeneous random walks in Rd, which may be recurrent in any dimension. The limit X is an elliptic martingale diffusion, which may be point-recurrent at the origin for any d 2. To characterize X, we introduce a (non-Euclidean) Riemannian metric on the unit sphere in Rd 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 X and thus develop the excursion theory of X without appealing to the strong Markov property. This leads to the uniqueness in law of the stochastic differential equation for X in Rd, whose coefficients are discontinuous at the origin. Using the Riemannian metric we can also detect whether the angular component of the excursions of X is time-reversible. If so, the excursions of X in Rd generalize the classical Pitman–Yor splitting-at-the-maximum property of Bessel excursions.

Citation

Georgiou, N., Mijatović, A., & Wade, A. R. (2019). Invariance principle for non-homogeneous random walks. Electronic Journal of Probability, 24, https://doi.org/10.1214/19-ejp302

Journal Article Type Article
Acceptance Date Mar 27, 2019
Online Publication Date May 18, 2019
Publication Date May 18, 2019
Deposit Date Jan 25, 2018
Publicly Available Date May 19, 2019
Journal Electronic Journal of Probability
Electronic ISSN 1083-6489
Publisher Institute of Mathematical Statistics
Peer Reviewed Peer Reviewed
Volume 24
DOI https://doi.org/10.1214/19-ejp302
Public URL https://durham-repository.worktribe.com/output/1336105

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