Stochastic Mappings and Random Distribution Fields II: Stationarity

Abstract

As a continuation of [3] this paper treats the stationary and stationarily cross-correlated multivariate stochastic mappings. Moreover for the case of multivariate random distribution fields, a particular form for the operator cross covariance distribution is given, from which a Kolmogorov type isomorphism theorem and a spectral representation of a stationary multivariate random distribution field are derived.

Stochastic mappings and random distribution fields: A correlation approach

“…random distribution fields, namely those having the property of operator stationarity. Such a study of stationarity, stationarily cross correlatedness, the corresponding spectral representations, as well as the periodically correlatedness are treated in [6] and [7]. An investigation of some other classes of m.s.o.…”

On Random Normal Operators and Their Spectral Measures

“…Definition 1. 3 We say that a random operator A ∈ R p ( ; G, H ) admits a random adjoint if there exists B ∈ R p ( ; H , G) such that for each x ∈ D(A) and y ∈ D(B) we have Ax(ω), y = x, By(ω) a.s.…”

“…In quite recent times, the study of stochastic processes or random fields was enlarged to the framework of multivariate stochastic mappings (see [2,20]) in order to treat in a unified way also other probabilistic concepts such as stochastic measures and stochastic integrals, random distributions or random distribution fields, as well as random operators (see [3][4][5]7,8,15,[17][18][19][20][21][22]), but also in an attempt to develop in this setting a corresponding random spectral theory (see [7,8,15,18,24,25]). B Pȃstorel Gaşpar pastorel.gaspar@uav.ro 1 Department of Mathematics and Computer Science, Faculty of Exact Sciences, "Aurel Vlaicu" University, Arad, Str.…”

“…For a study of such mappings see [19,20], for p = 0, 2 and [2][3][4] for p = 2. In this paper, we restrain ourselves to the case in which is a complex separable Hilbert space G (which can differ from H ) or some dense subspace of G. We shall call these p-order random operators.…”

On Selfadjoint Spectral Theory of Random Operators

“…In quite recent times the study of stochastic processes or random fields was enlarged to the framework of multivariate stochastic mappings (see [20], [3]) in order to treat in a unitary way also other probabilistic concepts such as stochastic measures and stochastic integrals, random distributions or random distribution fields, as well as random operators (see [19], [17], [20], [18], [23], [21], [22], [16], [4], [5], [6], [9], [8]), but also in an attempt to develop in this setting a corresponding random spectral theory (see [16], [24], [18], [9], [8]). Specifically, in [24], in terms of measurable families of deterministic continuous linear operators, the continuous normal and Hermitian random operators, random spectral measures were defined and the random version of the spectral (integral) representation theorems were given.…”