Some extremal problems for Gaussian and Empirical random fields.

Some important problems of probability and statistics can be reduced to the evaluation of supremum of some homogeneous functional defined on the Strasses ball in the space of smooth functions on the square. We give the solution of this extremal problem in two particular cases: when the functional is linear and continuous and when it is a superposition of two seminorms. As a result we obtain the rough large deviation asymptotic for Lp-norms of Brownian fields on the square, some Strassen type laws of iterated logarithm for functional of Brownian fields, and describe the conditions of local Bahadur optimality for some nonparametric independence tests such as generalized rank correlation coefficients.