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

Let ν ∈ M([0,∞[) be a xed probability measure. For each dimension p ∈ N, let (X n)n≥1 be i.i.d. R-valued random variables with radially symmetric distributions and radial distribution ν. We investigate the distribution of the euclidean length of S n := X p 1 + . . . + X p n for large parameters n and p. Depending on the growth of the dimension p = pn on the number of steps n we derive by the method of moments two complementary CLT's for the functional ‖S n‖2 with normal limits, namely for n/pn →∞ and n/pn → 0. Moreover, we present a CLT for the case n/pn → c ∈]0,∞[. Thereby we derive explicit formulas and asymptotic results for moments of radial distributed random variables on R. All limit theorems are considered also for orthogonal invariant random walks on the space Mp,q(R) of p× q matrices instead of R for p→∞ and some xed dimension q.