Limit theorems and exponential inequalities for canonical U- and V-statistics of dependent trials

Abstract

The limit behavior is studied for the distributions of normalized U- and V-statistics of an arbitrary order with canonical (degenerate) kernels, based on samples of increasing sizes from a stationary sequence of observations satisfying φ- or α-mixing. The case of m-dependent sequences is separately studied. The corresponding limit distributions are represented as infinite multilinear forms of a centered Gaussian sequence with a known covariance matrix. Moreover, under φ-mixing, exponential inequalities are obtained for the distribution tails of these statistics with bounded kernels.