Dr Nicholas Georgiou nicholas.georgiou@durham.ac.uk
Associate Professor
Invariance principle for non-homogeneous random walks
Georgiou, Nicholas; Mijatović, Aleksandar; Wade, Andrew R.
Authors
Aleksandar Mijatović
Professor Andrew Wade andrew.wade@durham.ac.uk
Professor
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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Publisher Licence URL
http://creativecommons.org/licenses/by/4.0/
Copyright Statement
This article was published under a Creative Commons Attribution 4.0 International License.
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