Central limit theorem for the empirical process

We discuss the functional central limit theorem (FCLT) for the empirical process of a moving-average stationary sequence with long memory. The cases of one-sided and double-sided moving averages are discussed. In the case of one-sided (causal) moving average, the FCLT is obtained under weak conditions of smoothness of the distribution and the existence of (2+δ)-moment of i.i.d. innovations, by using the martingale difference decomposition due to Ho and Hsing (1996, Ann. Statist. 24, 992–1014). In the case of double-sided moving average, the proof of the FCLT is based on an asymptotic expansion of the bivariate probability density.