  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. Freedman's 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 Freedman's 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 Schoenberg's result. Along the way, we
obtain a local field counterpart of an observation variously attributed to
Maxwell, Poincar\'e, 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 $n \to \infty$.
