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

Abstract

Let $\nu \in M^1([0,\infty [)$ be a fixed probability measure. For each dimension $p\in\b N$, let $(X_n^p{)}_{n\ge 1}$ be i.i.d. $\b R^p$-valued radial random variables with radial distribution $\nu$. We derive two central limit theorems for $\Vert X_1^p+...+X_n^p{\Vert }_2$ 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 $q$.