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

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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 > 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:	Date Event 18 May 2019 Published
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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