The slow convergence of multivariate U-statistic for nonstationary processes

The object is to study the asymptotic behavior of the multivariate U-statistic for nonstationary independent processes of a regular functional theta(F) where F is the distribution function of an observation. First, the asymptotic behavior of nonstationary independent processes is studied. Then the use of nonstationary dependent processes with a coefficient of mixing (absolute regularity rate beta (m) (mgreater than or equal to1)) is addressed. The convergence in law is established, under assumptions of uniform integrability of order q (>2) of the kernel symmetric functions for nonstationary independent observations and under the important assumption: the convergence of the nonstationary distributions functions for some norm.