2019/03/05 by Evans, Steven N., Raban, Daniel
#12J25 (Secondary) #60B99 #60G09 (Primary) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1903.02058
An infinite sequence of real random variables (ξ1, ξ2, …) is said to be rotatable if every finite subsequence (ξ1, …, ξn) has a spherically symmetric distribution. A celebrated theorem of Freedman states that (ξ1, ξ2, …) is rotatable if and only if ξj = τηj for all j, where (η1, η2, …) 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: ℝ+ → ℂ with ϕ(0) = 1 is completely monotone if and only if ϕn: ℝn → ℝ given by ϕn(x1, …, xn) = ϕ(x12 + ⋯ + xn2) is nonnegative definite for all n ∈ 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 (ζ1, …, ζn) is uniformly distributed on the sphere of radius √(n) in ℝn, then, for fixed k ∈ ℕ, the distribution of (ζ1, …, ζk) converges to that of a vector of k independent standard Gaussian random variables as n → ∞.