On a Class of Gaussian Distributions in Functional Space

А Д Егоров ,Anatoly Zherelo

doi:10.17713/ajs.v34i2.404

Abstract

An analog of reproducing Hilbert space of measure for a class of signed Gaussian distributions in the space of functionals is considered. Orthonormal bases in two special cases are constructed.

Highlights

In many problems concerning statistics of stochastic processes absolute continuity or singularity of distributions play important role
In Egorov (1997) an approach was developed to investigation of structure of the space L2(X, μ) over linear topological space X with nongaussian measure μ defined by characteristic functional
In present work we consider a class of nongaussian distributions in the space of nonlinear functionals associated with processes determined by signed measures defined on function spaces

Summary

Introduction

In many problems concerning statistics of stochastic processes absolute continuity or singularity of distributions play important role. In present work we consider a class of nongaussian distributions in the space of nonlinear functionals associated with processes determined by signed measures defined on function spaces. These measures are of unbounded variation but their characteristic functional have simple structure, that allows to carry out exact evaluations. Note that since we deal with functionals of stochastic process ξ(t) = ξ(t, ω), t ∈ T, ω ∈ (Ω, P ) it is convenient for us to use instead of P the corresponding distributions in functional space X of functions x(t), t ∈ T, with σ-algebra B generated by cylindrical sets In this case we can use the formula.

Gaussian Constructions Connected with Signed Measure

Orthonormal Basis in H

Now let