A Central Limit Theorem for a sequence of Brownian motions in the unit sphere in Rn

TL;DR

This paper establishes a Central Limit Theorem for sequences of Brownian motions on the unit sphere in R^n, and extends the results to n-dimensional Ornstein-Uhlenbeck processes, using stochastic differential equations.

Contribution

It introduces a CLT for spherical Brownian motions and generalizes it to Ornstein-Uhlenbeck processes in higher dimensions.

Findings

Proves a CLT for spherical Brownian motions.

Extends CLT results to Ornstein-Uhlenbeck processes.

Uses SDEs to derive the results.

Abstract

We use a Stochastic Differential Equation satisfied by Brownian motion taking values in the unit sphere $S_{n-1}subsetmathbb{R}^{n}$ and we obtain a Central Limit Theorem for a sequence of such Brownian motions. We also generalize the results to the case of the $n$-dimensional Ornstein-Uhlenbeck processes.