Quenched Invariance Principles for the Discrete Fourier Transforms of a Stationary Process

In this paper we study the asymptotic behavior of the normalized cadlag functions generated by the discrete Fourier transforms of a stationary centered squareintegrable process, started at a point. We prove that the quenched invariance principle holds for averaged frequencies under no assumption other than ergodicity, and that the result holds also for almost every xed frequency under a certain generalization of the Hannan condition and a certain rotated form of the Maxwell and Woodroofe condition, which is guaranteed for a.e. frequency under a condition of weak dependence that we specify. If the process is in particular weakly mixing, our results describe the asymptotic distributions of the normalized discrete Fourier transforms at every frequency other than 0 and π under the generalized Hannan condition. We prove also that under a certain regularity hypothesis the conditional centering is irrelevant for averaged frequencies, and that the same holds for a given xed frequency under the rotated Maxwell and Woodroofe condition but not necessarily under the generalized Hannan condition. In particular, we prove that the hypothesis of regularity is not su cient for functional convergence without random centering at a.e. xed frequency. The proofs are based on martingale approximations and combine results from Ergodic Theory of recent and classical origin with approximation results by con-