Central Limit Theorems for Radial Random Walks on $p\times q$ Matrices for $p\to\infty$

TL;DR

This paper establishes two central limit theorems for radial random walks in high-dimensional spaces and matrix spaces, revealing their asymptotic normality under different growth regimes of dimension and sample size.

Contribution

It introduces novel CLTs for high-dimensional radial and matrix-valued random walks, extending classical results to the regime where dimension grows large.

Findings

CLT for n much larger than p with known convergence rates

CLT for n much smaller than p using Bessel convolution asymptotics

Extension of CLTs to matrix spaces with $U(p)$-invariance

Abstract

Let $\nu\in M^1([0,\infty[)$ be a fixed probability measure. For each dimension $p\in\b N$, let $(X_n^p)_{n\ge1}$ be i.i.d. $\b R^p$-valued radial random variables with radial distribution $\nu$. We derive two central limit theorems for $ \|X_1^p+...+X_n^p\|_2$ for $n,p\to\infty$ with normal limits. The first CLT for $n>>p$ follows from known estimates of convergence in the CLT on $\b R^p$, while the second CLT for $n<<p$ will be a consequence of asymptotic properties of Bessel convolutions. Both limit theorems are considered also for $U(p)$-invariant random walks on the space of $p\times q$ matrices instead of $\b R^p$ for $p\to\infty$ and fixed dimension $q$.