The central limit theorem on nilpotent Lie groups

TL;DR

This paper proves a version of the central limit theorem for products of i.i.d. random variables on nilpotent Lie groups, revealing new phenomena due to bias and analyzing the limiting distributions and their properties.

Contribution

It introduces the first CLT for nilpotent Lie groups with bias, studies the limiting distributions, and characterizes when they are Gaussian, extending classical probability results.

Findings

New phenomena with bias: faster spread and limited support.

Established density estimates and support descriptions of the limit.

Proved Berry-Esseen bounds and an invariance principle for the distributions.

Abstract

We formulate and establish the central limit theorem for products of i.i.d. random variables on arbitrary simply connected nilpotent Lie groups, allowing a possible bias. Two new phenomena arise in the presence of a bias: (a) the walk spreads out at a higher rate in the ambient group, (b) the limiting hypoelliptic diffusion process may not have full support. We study the limiting distribution, prove estimates on its density and describe its support. We also establish corresponding Berry-Esseen bounds under optimal moment assumptions, as well as an analogue of Donsker's invariance principle. Various examples of nilpotent Lie groups are treated in detail showing the variety of different behaviours. We also obtain a characterization of when the limiting distribution is an ordinary gaussian and answer a question of Tutubalin from the 1960s regarding asymptotically close distributions on nilpotent Lie groups.