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

Let $\nu \in M^1([0,\mathrm{\infty }[)$ be a fixed probability measure. For each dimension $p\in \mathbb{N}$, let $(X_n^p{)}_{n\ge 1}$ 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 $S_n^p:=X_1^p+\mathrm{.}\mathrm{.}\mathrm{.}+X_n^p$ 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 $\mathrm{\mid }{S}_n^p{\mathrm{\mid }}_2$ with normal limits, namely for $n\mathrm{/}{p}_n\to \mathrm{\infty }$ and $n\mathrm{/}{p}_n\to 0$. Moreover, we present a CLT for the case $n\mathrm{/}{p}_n\to c\in ]0,\mathrm{\infty }[$. Thereby we derive explicit formulas and asymptotic results for moments of radial distributed random variables on $\text{b}{R}^p$. All limit theorems are considered also for orthogonal invariant random walks on the space $\text{b}{M}_{p,q}(\text{b}R)$ of $p\times q$ matrices instead of $\text{b}{R}^p$ for $p\to \mathrm{\infty }$ and some fixed dimension $q$.