Asymptotic Normality for a General Statistic from a Stationary Sequence

Abstract

Let $\{Z_i: -\infty < i < + \infty\}$ be a strictly stationary $\alpha$-mixing sequence. Without specifying the dependence model giving rise to $\{Z_i\}$, and without specifying the marginal distribution of $Z_i$, we address the question of asymptotic normality for a general statistic $t_n(Z_1,\ldots, Z_n)$. The main theoretical result is a set of necessary and sufficient conditions for joint asymptotic normality of $t_n$ and a subseries value $t_m (m \leq n)$. Our theorems on asymptotic normality are the natural analogs to earlier results that deal with general statistics from iid sequences, and to other results that apply to the sample mean from dependent sequences. Asymptotic normality of the sample mean and of the sample fractiles follows as a special case of our general statistic $t_n$.