Rotatable random sequences in local fields

An infinite sequence of real random variables $(xi_1, xi_2, dots)$ is said to be rotatable if every finite subsequence $(xi_1, dots, xi_n)$ has a spherically symmetric distribution. A celebrated theorem of Freedman states that $(xi_1, xi_2, dots)$ is rotatable if and only if $xi_j = tau eta_j$ for all $j$, where $(eta_1, eta_2, dots)$ is a sequence of independent standard Gaussian random variables and $tau$ is an independent nonnegative random variable. Freedmans theorem is equivalent to a classical result of Schoenberg which says that a continuous function $phi : mathbb{R}_+ to mathbb{C}$ with $phi(0) = 1$ is completely monotone if and only if $phi_n: mathbb{R}^n to mathbb{R}$ given by $phi_n(x_1, ldots, x_n) = phi(x_1^2 + cdots + x_n^2)$ is nonnegative definite for all $n in mathbb{N}$. We establish the analogue of Freedmans theorem for sequences of random variables taking values in local fields using probabilistic methods and then use it to establish a local field analogue of Schoenbergs result. Along the way, we obtain a local field counterpart of an observation variously attributed to Maxwell, Poincare, and Borel which says that if $(zeta_1, ldots, zeta_n)$ is uniformly distributed on the sphere of radius $sqrt{n}$ in $mathbb{R}^n$, then, for fixed $k in mathbb{N}$, the distribution of $(zeta_1, ldots, zeta_k)$ converges to that of a vector of $k$ independent standard Gaussian random variables as $n to infty$.