  The radial probability measures on $R^p$ are in a one-to-one correspondence
with probability measures on $[0,\infty[$ by taking images of measures w.r.t.
the Euclidean norm mapping. For fixed $\nu\in M^1([0,\infty[)$ and each
dimension p, we consider i.i.d. $R^p$-valued random variables $X_1^p,X_2^p,...$
with radial laws corresponding to $\nu$ as above. We derive weak and strong
laws of large numbers as well as a large deviation principle for the Euclidean
length processes $S_k^p:=\|X_1^p+...+X_k^p\|$ as k,p\to\infty in suitable ways.
  In fact, we derive these results in a higher rank setting, where $R^p$ is
replaced by the space of $p\times q$ matrices and $[0,\infty[$ by the cone
$\Pi_q$ of positive semidefinite matrices. Proofs are based on the fact that
the $(S_k^p)_{k\ge 0}$ form Markov chains on the cone whose transition
probabilities are given in terms Bessel functions $J_\mu$ of matrix argument
with an index $\mu$ depending on p. The limit theorems follow from new
asymptotic results for the $J_\mu$ as $\mu\to \infty$. Similar results are also
proven for certain Dunkl-type Bessel functions.
