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

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 $q$.