Random walks in Euclidean space

TL;DR

This paper studies the behavior of random walks generated by independent isometries in Euclidean space, proving local and global limit theorems under certain moment and non-degeneracy conditions.

Contribution

It establishes a local limit theorem and a broad-scale limit theorem for random walks in Euclidean space with new conditions and scale ranges.

Findings

Proves a local limit theorem under moment and non-degeneracy conditions.

Establishes a limit theorem on scales from e^(-cl^(1/4)) to l^(1/2).

Provides conditions for the convergence behavior of random isometric walks.

Abstract

Consider a sequence of independent random isometries of Euclidean space with a previously fixed probability law. Apply these isometries successively to the origin and consider the sequence of random points that we obtain this way. We prove a local limit theorem under a suitable moment condition and a necessary non-degeneracy condition. Under stronger hypothesis, we prove a limit theorem on a wide range of scales: between e^(-cl^(1/4)) and l^(1/2), where l is the number of steps.