Maxima of Gaussian Processes and Applications

• Definition of stochastic processes
• Gaussian distribution and Gaussian processes
• Regularity of the paths
• Basis inequalities: Plackett-Slepian, Borell Sudakov Tsirelson
• Rice formulas for the number of crossings of a process. Gaussian and non Gaussian case
• Basis bounds for the distribution of the maximum of a Gaussian process
• Application to change points detection
• Application to gene detection
• Computation of Gaussian integrals in high dimension by Monte-Carlo Quasi-Monte-Carlo (MCQMC) method
• Application to the prediction of electrical load curves
• Rice formula for random fields
• Application to the record method
• Application to roots of random polynomials

Based on take-home exam

3 ECTS for all tracks

Course outline