Limit theorems for non-linear functionals of stationary Gaussian random fields
Abstract
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 this note first I discuss the spectral representation of the covariance function of a Gaussian stationary rendom field by means of the spectral measure and the representation of the elements of the random field by means of a random integral with respect to the random spectral measure. Then I construct the multiple random integrals with respect to the random spectral measure and prove their most important properties. Finally I show some interesting applications of these multiple random integrals. In particular, I prove some non-trivial non-Gaussian limit theorems
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2017
- DOI:
- 10.48550/arXiv.1708.03313
- arXiv:
- arXiv:1708.03313
- Bibcode:
- 2017arXiv170803313M
- Keywords:
-
- Mathematics - Probability;
- primary: 60G60;
- secondary: 60H05;
- 60F99
- E-Print:
- 82 pages, 3 figures, the text of a series of lectures