M.Sc Thesis | |

M.Sc Student | Naitzat Gregory |
---|---|

Subject | A Central Limit Theorem for the Euler Integral of Gaussian Random Fields |

Department | Department of Electrical and Computer Engineering |

Supervisor | PROFESSOR EMERITUS Robert Adler |

Full Thesis text |

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 *T _{n}
= [0,m]^{n}* converges in distribution to a Gaussian random
variable as