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Invariance principle for non-homogeneous random walks.

Georgiou, Nicholas and Mijatović, Aleksandar and Wade, Andrew R. (2019) 'Invariance principle for non-homogeneous random walks.', Electronic journal of probability., 24 . p. 48.


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.

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Publisher statement:This article was published under a Creative Commons Attribution 4.0 International License.
Date accepted:27 March 2019
Date deposited:10 April 2019
Date of first online publication:18 May 2019
Date first made open access:19 May 2019

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