On almost sure representations for long memory sequences

摘要

Let $G(cdot)$ be a Borel function applied to a stationary long memory sequence ${ X_i }$ of standard Gaussian random variables. Focusing on the process ${G(X_i)}$, the present paper establishes the almost sure representation for the empirical quantile process, that is, Bahadur's representation, and for the empirical process with respect to sample mean. Statistical applications of the representations are also addressed.