Asymptotic linearity and limit distributions, approximations

Linear and quadratic forms as well as other low degree polynomials play an important role in statistical inference. Asymptotic results and limit distributions are obtained for a class of statistics depending on mu + X, with X any random vector and mu non-random vector with parallel to mu parallel to -> + infinity. This class contain the polynomials in mu + X. An application to the case of normal X is presented. This application includes a new central limit theorem which is connected with the increase of non-centrality for samples of fixed size. Moreover upper bounds for the suprema of the differences between exact and approximate distributions and their quantiles are obtained. (C) 2009 Elsevier B.V. All rights reserved.