Contributions to Central Limit Theory for Dependent Variables

• Published in: The Annals of mathematical statistics 1968-08, Vol.39 (4), p.1158-1175
• Main Author: Serfling, R. J.
• Format: Article
• Language: English
• Subjects: Central limit theorem; Ergodic theory; Integers; Logical givens; Martingales; Mathematical moments; Mathematical theorems; Mathematics; Probability distributions; Random variables
• ISSN: 0003-4851; 2168-8990
• DOI: 10.1214/aoms/1177698240

Considerations on stochastic models frequently involve sums of dependent random variables (rv's). In many such cases, it is worthwhile to know if asymptotic normality holds. If so, inference might be put on a nonparametric basis, or the asymptotic properties of a test might become more easily evaluated for certain alternatives. Of particular interest, for example, is the question of when a weakly stationary sequence of rv's possesses the central limit property, by which is meant that the sum ∑n 1Xi, suitably normed, is asymptotically normal in distribution. The feeling of many experimenters that the normal approximation is valid in situations "where a stationary process has been observed during a time interval long compared to time lags for which correlation is appreciable" has been discussed by Grenander and Rosenblatt ([10]; 181). (See Section 5 for definitions of stationarity.) The general class of sequences {Xi}-∞ ∞considered in this paper is that whose members satisfy the variance condition \begin{equation*}\tag{1.1}\operatorname{Var} (\sum^{a+n}_{a+1} X_i) \sim nA^2\text{uniformly in} a (n \rightarrow \infty) (A^2 > 0).\end{equation*} Included in this class are the weakly stationary sequences for which the covariances rjhave convergent sum ∑1 ∞rj. A familiar example is a sequence of mutually orthogonal rv's having common mean and common variance. As a mathematical convenience, it shall be assumed (without loss of generality) that the sequences {Xi} under consideration satisfy$E(X_i) \equiv 0$, for the sequences {Xi} and {Xi- E(Xi)} are interchangeable as far as concerns the question of asymptotic normality under the assumption (1.1). As a practical convenience, it shall be assumed for each sequence {Xi} that the absolute central moments E|Xi- E(Xi)|νare bounded uniformly in i for some$\nu > 2$(ν may depend upon the sequence). When (1.1) holds, this is a mild additional restriction and a typical criterion for verifying a Lindeberg restriction ([15]; 295). We shall therefore confine attention to sequences {Xi} which satisfy the following basic assumptions (A): \begin{equation*}\tag{A1}E(X_i) \equiv 0,\end{equation*}\begin{equation*}\tag{A2}E(T_a^2) \sim A^2 \text{uniformly in} a (n \rightarrow \infty) (A^2 > 0),\end{equation*}\begin{equation*}\tag{A3}E|X_i|^{2+\delta} \leqq M (\text{for some} \delta > 0 \text{and} M < \infty),\end{equation*} where Tadenotes the normed sum n-1/2∑a+n a+1Xi. Note that the formulations of (A2) and (A3) presuppose (A1). We