Limit theorems for projections of random walk on a hypersphere

TL;DR

This paper proves that one-dimensional projections of high-dimensional random walks on hyperspheres converge to an Ornstein-Uhlenbeck process, extending classical results like Poincaré's and Bernoulli-Laplace CLT to a functional setting.

Contribution

It establishes a weak convergence of projections of scaled random walks on hyperspheres to Ornstein-Uhlenbeck processes, generalizing classical limit theorems to a functional context.

Findings

Projections of random walks on hyperspheres converge to Ornstein-Uhlenbeck processes.

The same convergence holds for spherical Brownian motion.

Results suggest a new functional perspective on classical high-dimensional limit theorems.

Abstract

We show that almost any one-dimensional projection of a suitably scaled random walk on a hypercube, inscribed in a hypersphere, converges weakly to an Ornstein-Uhlenbeck process as the dimension of the sphere tends to infinity. We also observe that the same result holds when the random walk is replaced with spherical Brownian motion. This latter result can be viewed as a "functional" generalisation of Poincar\'e's observation for projections of uniform measure on high dimensional spheres; the former result is an analogous generalisation of the Bernoulli-Laplace central limit theorem. Given the relation of these two classic results to the central limit theorem for convex bodies, the modest results provided here would appear to motivate a functional generalisation.