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. 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 $n \to \infty$.