Limit Theorems for Random Walks in the Hyperbolic Space

TL;DR

This paper establishes central, local limit theorems and a law of large numbers for random walks in hyperbolic space, using the ball model and Möbius transformations to analyze their asymptotic behavior.

Contribution

It introduces new limit theorems for random walks in hyperbolic space using the ball model and Möbius addition, extending classical probabilistic results to non-Euclidean geometry.

Findings

Proved central limit theorem for hyperbolic random walks

Established local limit theorem in hyperbolic space

Derived a law of large numbers for the walks

Abstract

We prove central and local limit theorems for random walks on the Poincar{\'e} hyperbolic space of dimension n {\v e} 2. To this end we use the ball model and describe the walk therein through the M{\"o}bius addition and multiplication. This also allows to derive a corresponding law of large numbers.