A potpourri of results in the general theory of stochastic noises

TL;DR

This paper explores the spectral structure of noise Boolean algebras, providing explicit descriptions of their spectra, conditions for product-typeness, and new characterizations of classical versus non-classical noise.

Contribution

It introduces spectral decompositions for noise Boolean algebras, characterizes classicality and blackness via spectral independence, and develops noise projections and tensor structures in the spectral space.

Findings

Explicit spectral descriptions of noise Boolean algebras

Discreteness of spectral measures implies product-typeness

New criteria for classicality and blackness based on spectral independence

Abstract

The objects under inspection, on a given probability space, are noise(-type) Boolean algebras -- distributive non-empty sublattices of the lattice of all complete sub-$\sigma$-fields, whose every element admits an independent complement. Special attention is given to the spectral decompositions of the algebras of operators generated by the conditional expectations of their members (acting on $\mathrm{L}^2$). Atoms of the spectra are identified in explicit terms. For a reverse filtration admitting an innovating sequence of equiprobable random signs, a discreteness property of the spectral measure of the associated noise Boolean algebra is shown to imply product-typeness. Noise projections on the spectral space are introduced, which correspond to restricting a noise to a part of its domain space. They appear to play a natural (albeit technical) r\^ole in the general analysis. In particular through them manifests itself the tensor structure of the spectral decomposition. The spectrum is precisely delineated in the classical case (i.e. when the noise Boolean algebra is complete), a kind-of standard "symmetric Fock space" form thereof is procured. The latter result leads to a new characterization of classicality and blackness involving "spectral independence".