An infinite-dimensional helix invariant under spherical projections

Electronic Communications in Probability

Zakhar Kabluchko¹ 

关键词: Gaussian process;    curve in Hilbert space;    spherical projection;    metric space;    triangle equality;    zeroes;    Pfaffian point process;    determinantal point process;   

DOI  :  10.1214/19-ECP234

学科分类：统计和概率

来源: Institute of Mathematical Statistics

【 摘 要 】

We classify all subsets $S$ of the projective Hilbert space with the following property: for every point $\pm s_{0}\in S$, the spherical projection of $S\backslash \{\pm s_{0}\}$ on the hyperplane orthogonal to $\pm s_{0}$ is isometric to $S\backslash \{\pm s_{0}\}$. In probabilistic terms, this means that we characterize all zero-mean Gaussian processes $Z=(Z(t))_{t\in T}$ with the property that for every $s_{0}\in T$ the conditional distribution of $(Z(t))_{t\in T}$ given that $Z(s_{0})=0$ coincides with the distribution of $(\varphi (t; s_{0}) Z(t))_{t\in T}$ for some function $\varphi (t;s_{0})$. A basic example of such process is the stationary zero-mean Gaussian process $(X(t))_{t\in \mathbb{R} }$ with covariance function $\mathbb E [X(s) X(t)] = 1/\cosh (t-s)$. We show that, in general, the process $Z$ can be decomposed into a union of mutually independent processes of two types: (i) processes of the form $(a(t) X(\psi (t)))_{t\in T}$, with $a: T\to \mathbb{R} $, $\psi (t): T\to \mathbb{R} $, and (ii) certain exceptional Gaussian processes defined on four-point index sets. The above problem is reduced to the classification of metric spaces in which in every triangle the largest side equals the sum of the remaining two sides.

【 授权许可】

CC BY