M.Sc Thesis

M.Sc StudentNaitzat Gregory
SubjectA Central Limit Theorem for the Euler Integral of Gaussian
Random Fields
DepartmentDepartment of Electrical and Computer Engineering
Supervisor PROFESSOR EMERITUS Robert Adler
Full Thesis textFull thesis text - English Version


On the one hand, applied topology is a new field of study, the purpose of which is to develop topological and related concepts for data analysis and other "real-world" problems. On the other hand, for some time

probabilists have been using topological notions as a means to study the properties of random fields, in particular, the properties of the Euler characteristic. This connection between probability and topology is advantageous for both fields, and allows the further expansion of applied topology, increasing its ability to address the uncertainties of the "real world".

In this study, we look at the Euler integral. The Euler integral is a good example of the kind of tools used by applied topology and is a relatively old concept built on the topological notion of Euler-Poincare characteristics. It exploits the inclusion-exclusion property of the Euler characteristic to define an integration-like operation on functions on topological spaces. In recent years, Euler integration has experienced rapid development, not least because of its novel application in the target enumeration problem, for which the current thesis has direct consequences.  We formulate, for the first time, the central limit theorem (CLT) for the Euler integral of random Gaussian fields of arbitrary dimension n. In particular, under suitable conditions, we show that the normalized Euler integral of a random (tame) Gaussian field on a n-dimensional cube Tn = [0,m]n  converges in distribution to a Gaussian random variable as m goes to infinity.  Our approach to the proof of the CLT follows very closely that in the recent work of Estrade and Leon in which a CLT for the Euler characteristic of the excursion sets of random Gaussian fields is proven.  The construction of the proof is based on the representation of the related random processes via a Wiener chaos expansion, and then an application of a general CLT developed by Nourdin and Peccati.  This general CLT is a powerful result that originates from Stein's method and Malliavin calculus.  A direct calculation of the mean value of the Euler integral of an isotropic (tame) Gaussian field concludes this thesis. We show its dependence on only the one-dimensional measure of the volume of integration, thus independently showing its surprising scaling property, which was first discovered by Bobrowski and Borman.