Moscow Mathematical Journal
Volume 20, Issue 4, October–December 2020 pp. 749–812.
The Theory of Wiener–Itô Integrals in Vector Valued Gaussian Stationary Random Fields. Part I
The subject of this work is the multivariate
generalization of the theory of multiple Wiener–Itŏ integrals. In
the scalar valued case this theory was described by the author in 2014. The proofs of the present
paper apply the technique of that work, but in the proof of some
results new ideas were needed. The motivation for this study was a
result in the paper “Limit theorems for
nonlinear functionals of a stationary Gaussian sequence of vectors”
(1994) by Arcones, which contained the multivariate
generalization of a non-central limit theorem for non-linear
functionals of Gaussian stationary random fields presented in
a paper by R.L. Dobrushin and the
author. However, the formulation of Arcones’ result was
incorrect. To present it in a correct form, the
multivariate version of the theory explained in my
work of 2014 has to be worked out, because the notions introduced
in this theory are needed in its formulation. This is done in the
present paper. In its continuation it
will be explained how to work out a method with the help
of the results in this work that enables us to prove non-Gaussian
limit theorems for non-linear functionals of vector valued Gaussian
stationary random fields. The right version of Arcones’ result
presented also in the introduction of this work will be formulated and
proved with its help in a future paper of mine. 2010 Math. Subj. Class. 60G10, 60G15, 60H05.
Authors:
Péter Major (1)
Author institution:(1) Alfréd Rényi Institute of Mathematics, Budapest, P.O.B. 127 H-1364, Hungary
Summary:
Keywords: Vector valued Gaussian stationary random fields,
multivariate spectral measure, multivariate random spectral measure, multiple
integral with respect to random spectral measure, diagram formula.
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