Rotatable random sequences in local fields

Abstract

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, Poincaré, 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 $.