The principal investigator will continue his study of the upper and lower tail behavior of vector valued Brownian motion. Results in this direction can be used for estimating the behavior of other Gaussian processes such as a two parameter Kiefer process. The oscillatory behavior of these processes is studied via the discrepancy between collections of sample paths and certain non-random set functions. The non-random set functions are metric entropy of balls in the reproducing kernel Hilbert space of the Brownian motion. Error rates for the convergence of the processes to their cluster sets and the approximate density of sample path collections is established. The research will contribute to the observable regularities within a widely-studied framework involving long-time chaos.
