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Series of problems in Probability Theory
This is the (extended) text 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. (These proofs 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 correlation 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 the methods of the Wiener--Ito integrals and
of the Fourier analysis are unified. 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 are the limit fields 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 part 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.
78 pages