{ "id": "1708.03313", "version": "v1", "published": "2017-08-10T17:46:58.000Z", "updated": "2017-08-10T17:46:58.000Z", "title": "Limit theorems for non-linear functionals of stationary Gaussian random fields", "authors": [ "Peter Major" ], "comment": "82 pages, 3 figures, the text of a series of lectures", "categories": [ "math.PR" ], "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", "revisions": [ { "version": "v1", "updated": "2017-08-10T17:46:58.000Z" } ], "analyses": { "subjects": [ "60G60", "60H05", "60F99" ], "keywords": [ "stationary gaussian random fields", "non-linear functionals", "note multiple wiener-ito integrals", "lecture note multiple wiener-ito", "multiple random integrals" ], "tags": [ "lecture notes" ], "note": { "typesetting": "TeX", "pages": 82, "language": "en", "license": "arXiv", "status": "editable" } } }