Invariance principle for non-homogeneous random walks

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

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