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Invariance principle for non-homogeneous random walks

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Georgiou, Nicholas, Mijatović, Aleksandar and Wade, Andrew R. (2019) Invariance principle for non-homogeneous random walks. Electronic Journal of Probability, 24 . 48. doi:10.1214/19-EJP302 ISSN 1083-6489.

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Official URL: http://dx.doi.org/10.1214/19-EJP302

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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.

Item Type: Journal Article
Divisions: Faculty of Science, Engineering and Medicine > Science > Statistics
Journal or Publication Title: Electronic Journal of Probability
Publisher: University of Washington. Dept. of Mathematics
ISSN: 1083-6489
Official Date: 18 May 2019
Dates:
DateEvent
18 May 2019Published
Volume: 24
Article Number: 48
DOI: 10.1214/19-EJP302
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access

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