High extrema of Gaussian chaos processes

摘要

Let xi ( t)=(xi (1)(t),aEuro broken vertical bar,xi (d) (t)) be a Gaussian stationary vector process. Let be a homogeneous function. We study probabilities of large extrema of the Gaussian chaos process g(xi(t)). Important examples include and . We review existing results partially obtained in collaboration with E. Hashorva, D. Korshunov, and A. Zhdanov. We also present the principal methods of our investigations which are the Laplace asymptotic method and other asymptotic methods for probabilities of high excursions of Gaussian vector process' trajectories.