On the central limit theorem for negatively correlated random variables with negatively correlated squares.
The main result of the paper is as follows. Let \(\{X_{nj}, 1\leq j\leq k_n\}\) be a double array of random variables with zero mean such that \(\lim \sup _{n\to \infty } s_n^{-4}\sum _{i\neq j} \text{Cov} (X_{ni}^2, X_{nj}^2) \leq 0\) where \(s_n^2 = \sum _{j=1}^{k_n} \text{Var} (X_{nj}),\) and such that the Lindeberg condition is satisfied. Let \(Z_n = s_n^{-1}(X_{n1}+\dots + X_{nk_n}) \) be uniformly integrable. It is proved by using the Stein method that \(Z_n\) has asymptotically standard normal distribution if and only if \[ \lim _{n\to \infty }s_n^{-1}\sum _{j=1}^{k_n} E[X_{nj} \exp \{it Z_{nj}\}] = 0, \] where \(Z_{nj} = Z_n - s_n^{-1}X_{nj}.\) The asymptotic normality for negatively correlated random variables with negatively correlated squares then follows as a corollary.
scientific article; zbMATH DE number 2098080