Limit theorems for radial random walks on Euclidean spaces of high dimensions

TL;DR

This paper establishes limit theorems for the Euclidean lengths of high-dimensional radial random walks, revealing different Gaussian behaviors depending on the growth rate of dimension relative to the number of steps.

Contribution

It derives new central limit theorems for radial sums in high dimensions, covering various growth regimes of dimension relative to steps, and extends results to matrix spaces.

Findings

CLTs for $|S_n^p|_2$ with different regimes of $n/p_n$

Explicit formulas for moments of radial distributions

Extension of limit theorems to matrix-valued random walks

Abstract

Let $\nu\in M^1([0,\infty[)$ be a fixed probability measure. For each dimension $p\in \mathbb{N}$, let $(X_n^{p})_{n\geq1}$ be i.i.d. $\mathbb{R}^p$-valued random variables with radially symmetric distributions and radial distribution $\nu$. We investigate the distribution of the Euclidean length of $S_n^{p}:=X_1^{p}+...+ X_n^{p}$ for large parameters $n$ and $p$. Depending on the growth of the dimension $p=p_n$ we derive by the method of moments two complementary CLT's for the functional $|S_n^{p}|_2$ with normal limits, namely for $n/p_n \to \infty$ and $n/p_n \to 0$. Moreover, we present a CLT for the case $n/p_n \to c\in]0,\infty[$. Thereby we derive explicit formulas and asymptotic results for moments of radial distributed random variables on $\b R^p$. All limit theorems are considered also for orthogonal invariant random walks on the space $\b M_{p,q}(\b R)$ of $p\times q$ matrices instead of $\b R^p$ for $p\to \infty$ and some fixed dimension $q$.