Invariance principle for non-homogeneous random walks.

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.