Rotatable random sequences in local fields

Abstract

An infinite sequence of real random variables $(ξ_1, ξ_2, \dots)$ is said to be rotatable if every finite subsequence $(ξ_1, \dots, ξ_n)$ has a spherically symmetric distribution. A celebrated theorem of Freedman states that $(ξ_1, ξ_2, \dots)$ is rotatable if and only if $ξ_j = τη_j$ for all $j$, where $(η_1, η_2, \dots)$ is a sequence of independent standard Gaussian random variables and $τ$ is an independent nonnegative random variable. Freedman's theorem is equivalent to a classical result of Schoenberg which says that a continuous function $ϕ: \mathbb{R}_+ \to \mathbb{C}$ with $ϕ(0) = 1$ is completely monotone if and only if $ϕ_n: \mathbb{R}^n \to \mathbb{R}$ given by $ϕ_n(x_1, \ldots, x_n) = ϕ(x_1^2 + \cdots + x_n^2)$ 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, Poincaré, and Borel which says that if $(ζ_1, \ldots, ζ_n)$ is uniformly distributed on the sphere of radius $\sqrt{n}$ in $\mathbb{R}^n$, then, for fixed $k \in \mathbb{N}$, the distribution of $(ζ_1, \ldots, ζ_k)$ converges to that of a vector of $k$ independent standard Gaussian random variables as $n \to \infty$.