On the central limit theorem for negatively correlated random variables with negatively correlated squares

Using Steinʹs method, assuming Lindebergʹs condition, we find a necessary and sufficient condition for the central limit theorem to hold for an array of random variables such that the variables in each row are negatively correlated (i.e., every pair has negative covariance) and their squares are also negatively correlated (in fact, a somewhat more general result is shown). In particular, we obtain a necessary and sufficient condition for the central limit theorem to hold for an array of pairwise independent random variables satisfying Lindebergʹs condition. A collection of random variables is said to be jointly symmetric if finite-dimensional joint distributions do not change when a subset of the variables is multiplied by −1. A corollary of our main result is that the central limit theorem holds for pairwise independent jointly symmetric random variables under Lindebergʹs condition. We also prove a central limit theorem for a triangular array of variables satisfying some size constraints and where the n variables in each row are φ(n)-tuplewise independent, i.e., every subset of cardinality no greater than φ(n) is independent, where φ is a function such that φ(n)/n1/2→∞.