An infinite-dimensional helix invariant under spherical projections

Abstract

We classify all subsets S of the projective Hilbert space with the following property: for every point ±s 0 ∈ S, the spherical projection of S\{±s 0 } to the hyperplane orthogonal to ±s 0 is isometric to S\{±s 0 }. In probabilistic terms, this means that we characterize all zero-mean Gaussian processes Z = (Z(t)) t∈T with the property that for every s 0 ∈ T the conditional distribution of (Z(t)) t∈T given that Z(s 0 ) = 0 coincides with the distribution of (ϕ(t; s 0 )Z(t)) t∈T for some function ϕ(t; s 0 ). A basic example of such process is the stationary zero-mean Gaussian process (X(t)) t∈R with covariance function 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(ψ(t))) t∈T , with a : T → R, ψ(t) : T → 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.

Limit theorems for discounted convergent perpetuities

“…Such a process has appeared in the recent articles [8], [17] and [18]. The latter paper provides additional references.…”

Limit theorems for random Dirichlet series

“…In particular, this means that the process (S α (t)) t∈R is a complex-valued stationary Gaussian process on R. Setting α = 0, replacing I (0; e 2z ) with I 1 (0; e 2z ) in the definition of S α and only considering z ∈ R we obtain a centered real-valued stationary Gaussian process S 0 , which was recently investigated in [12]. Another important time-change is constructed as follows.…”