Compact convergence of σ-fields and relaxed conditional expectation

Artstein, Zvi

doi:10.1007/pl00008787

Cited by 10 publications

(22 citation statements)

References 22 publications

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“…Of course {K = 0} = S 0 P a.e.-µ and H (0) is the one-dimensional space of constants. F stb := F stb (B) := σ(H (1) ) = σ(H stb ) [36, Proposition 7.9] is the stable σ-field. The first chaos, H (1) = {f ∈ L 2 (P) : Proposition 7.8], is especially important; its element are the square-integrable additive integrals of B, and B is said to be classical/linearizable/white (resp.…”

Section: (C) For a Setmentioning

confidence: 99%

“…F stb = 0 P , but 0 P = 1 P ). If B is classical, then x = σ(L 2 (P| x ) ∩ H (1) ) for x ∈ B [36, Lemma 6.2]. We put H sens := H sens (B) := H({K = ∞}).…”

Section: (C) For a Setmentioning

confidence: 99%

“…Existence of atoms precludes blackness: if a ∈ B is an atom of the Boolean algebra B and 0 P = 1 P , then {0} = L 2 (P| a ) 0 ⊂ H (1) . If B 0 is a noise Boolean algebra with the same closure as B (in particular if B 0 is dense in B), then B 0 is black (resp.…”

Section: (C) For a Setmentioning

confidence: 99%

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A potpourri of results in the general theory of stochastic noises

Vidmar¹

2021

Preprint

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The objects under inspection, on a given probability space, are noise(-type) Boolean algebrasdistributive non-empty sublattices of the lattice of all complete sub-σ-fields, whose every element admits an independent complement. Special attention is given to the spectral decompositions of the algebras of operators (acting on L 2 ) generated by the conditional expectations w.r.t. the members of said noise Boolean algebras.Atoms of the spectrum 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) rôle 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". Contents 1. Introduction 2. Basic notions, setting, notation, preliminaries 2.1. Generalities 2.2. Noise Boolean algebras 2.3. Spectrum and chaos decomposition 3. Some Bonferroni-type inequalities 4. Topology of noises 4.1. (Non)compactness and existence of limits 4.2. Various notions of completeness 4.3. Convergence and continuity in a classical noise Boolean algebra 4.4. Miscellany 5. Atoms of the spectral measure 5.1. General considerations: identification of the spectral atoms 5.2. Study of a class of examples: reverse filtrations and their innovations 6. Interactions between restrictions/projections of the spectrum and subnoises

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Section: (C) For a Setmentioning

confidence: 99%

“…F stb = 0 P , but 0 P = 1 P ). If B is classical, then x = σ(L 2 (P| x ) ∩ H (1) ) for x ∈ B [36, Lemma 6.2]. We put H sens := H sens (B) := H({K = ∞}).…”

Section: (C) For a Setmentioning

confidence: 99%

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A potpourri of results in the general theory of stochastic noises

Vidmar¹

2021

Preprint

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“…(Ω, S, P; E) of integrable Young measures on T. Multivalued extensions which keep some of the properties of these operations are constructed in [Art01a,Art01b,AP03].…”

Section: Integrable Young Measures and L P E Spacesmentioning

confidence: 99%

Young Measures on Topological Spaces

Castaing¹,

Fitte²,

Valadier³

2004

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“…; Kudo (1974) and Rogge (1974) investigated the convergence of σ-algebras in connection with problems arising in probability and mathematical statistics, and there has been extensive work in mathematical economics on the formulation of topologies on the space of information represented as equivalence classes of sub-σ-algebras of an abstract probability space that draws on their investigations; see Allen (1983Allen ( , 1986; Cotter (1986Cotter ( , 1987Cotter ( , 1991; Stinchcombe (1989Stinchcombe ( , 1993; Monderer and Samet (1996); Artstein (2001); Van Zandt (2002); Correia-da-Silva and Hervés-Beloso (2007); Einy et al (2005) and their references.…”

Section: Introductionmentioning

confidence: 99%