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 $\varphi :{\mathbb{R}}_{+}\to \mathbb{C}$ with $\varphi (0)=1$ is completely monotone if and only if ${\varphi }_n:{\mathbb{R}}^n\to \mathbb{R}$ given by ${\varphi }_n(x_1,\dots ,x_n)=\varphi (x_1^2+\cdots +x_n^2)$ 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 $({\zeta }_1,\dots ,{\zeta }_n)$ is uniformly distributed on the sphere of radius $\sqrt{n}$ in ${\mathbb{R}}^n$, then, for fixed $k\in \mathbb{N}$, the distribution of $({\zeta }_1,\dots ,{\zeta }_k)$ converges to that of a vector of $k$ independent standard Gaussian random variables as $n\to \infty$.

arXiv comment13 pages