Contributions to Central Limit Theory for Dependent Variables
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 $\sum^n_1 X_i$, 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 (; 181). (See Section 5 for definitions of stationarity.)
Serfling, R. J. (1968). Contributions to Central Limit Theory for Dependent Variables. Annals of Mathematical Statistics, 39(4), 1158-1175.