Asymptotic normality of quadratic forms with random vectors of increasing dimension

This paper provides sufficient conditions for the asymptotic normality of quadratic forms of averages of random vectors of increasing dimension and improves on conditions found in the literature. Such results are needed in applications of Owen's empirical likelihood when the number of constraints is allowed to grow with the sample size. Indeed, the results of this paper are already used in Peng and Schick (2013) for this purpose. We also demonstrate how our results can be used to obtain the asymptotic distribution of the empirical likelihood with an increasing number of constraints under contiguous alternatives. In addition, we discuss potential applications of our result. The first example focuses on a chi-square test with an increasing number of cells. The second example treats testing for the equality of the marginal distributions of a bivariate random vector. The third example generalizes a result of Schott (2005) by showing that a standardized version of his test for diagonality of the dispersion matrix of a normal random vector is asymptotically standard normal even if the dimension increases faster than the sample size. Schott's result requires the dimension and the sample size to be of the same order.