  Some problems of statistics can be reduced to extremal problems of minimizing
functionals of smooth functions defined on the cube $[0,1]^m$, $m\geq 2$. In
this paper, we study a class of extremal problems that is closely connected to
the problem of testing multivariate independence. By solving the extremal
problem, we provide a unified approach to establishing weak convergence for a
wide class of empirical processes which emerge in connection with testing
independence. The use of our result is also illustrated by describing the
domain of local asymptotic optimality of some nonparametric tests of
independence.
