Limit theorems for non-linear functionals of stationary Gaussian random fields

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    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.

    82 pages