Back to the
Series of problems in Probability Theory
This is an extended version of a series of talks I held at the
University of Bochum in 2017 about limit theorems for non-linear
functionals of stationary Gaussian random fields.
The goal of these talks was to give a fairly detailed introduction to
the theory leading to such results, even if some of the results are
presented without proof. On the other hand, I gave a simpler
proof for some of the results. (The proofs omitted from this
text can be found in my Springer Lecture Note
Multiple Wiener--Ito Integrals).
In the introduction I formulate the basic problem of this lecture,
and introduce some important notions. In the second section I
explain the representation of the covariance function of a
stationary Gaussian field as the Fourier transform of its spectral
measure, and then I show that a so-called random spectral measure
can be constructed, and the elements of the stationary random field
can be represented as its random Fourier transforms. An important
point of this discussion is the introduction of generalized
stationary random fields. Beside the proof of their most important
properties, the motivation for their introduction is also discussed.
In the third section the multiple Wiener--Ito integrals are
constructed, and the motivation for their introduction is explained.
More precisely, a version of this notion, introduced by R. L. Dobrushin
is explained, where we integrate with respect to a random spectral
measure. This enables us to apply the methods of the Wiener--Ito
integrals and Fourier analysis simultaneously. In the fourth
section I discuss the two most important results about
Wiener--Ito integrals, the diagram formula about the calculation
of the product of Wiener--Ito integrals and Ito's formula.
Section 5 contains some important applications of these results.
The canonical representation of the so-called subordinated
stationary fields is given, and self-similar random fields are
constructed. The self-similar random fields appear as the limit
in limit theorems for sums of strongly dependent random variables.
Besides, I present some non-trivial estimates on the
tail-distribution of Wiener--Ito integrals. In the last section
of this note, in section 6 I present some non-trivial
(non-Gaussian) limit theorems for non-linear functionals of a
stationary Gaussian fields. The proof applies the results of
the previous sections.
I finish this text with an Appendix, where I present the proof
of a technical result needed in the construction of Wiener--Ito
integrals. My Lecture Note contains a rather complicated
proof of this result. I hope that I could present here a more
transparent and understandable proof.