Electronic Communications in Probability

Rotatable random sequences in local fields

Steven N. Evans and Daniel Raban

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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 $.

Article information

Electron. Commun. Probab., Volume 24 (2019), paper no. 37, 12 pp.

Received: 6 March 2019
Accepted: 3 April 2019
First available in Project Euclid: 22 June 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B99: None of the above, but in this section 60G09: Exchangeability
Secondary: 12J25: Non-Archimedean valued fields [See also 30G06, 32P05, 46S10, 47S10]

$p$-adic $p$-series exchangeable spherically symmetric Gaussian total variation completely monotone nonnegative definite

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Evans, Steven N.; Raban, Daniel. Rotatable random sequences in local fields. Electron. Commun. Probab. 24 (2019), paper no. 37, 12 pp. doi:10.1214/19-ECP227. https://projecteuclid.org/euclid.ecp/1561169053

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