From nonstationarity to stationarity via $1/f$ noise: discrete Fourier transforms and sample mean asymptotics for testing

Abstract

We study the asymptotic behaviour of different statistics for time series exhibiting long memory and nonstationarity. For processes with memory parameter $d\in(-1/2,3/2)$, we derive the joint limiting distribution of discrete Fourier transforms at a fixed number of Fourier frequencies, with a unified normalization. The resulting limits are Gaussian with an explicit covariance structure. Particular attention is given to the boundary case $d=1/2$, also known as $1/f$ noise. We show that logarithmic corrections yield nondegenerate limits for sample mean and sample variance leading to explicit asymptotic distributions of $χ^2$ type. We construct a statistic that combines the sample mean, the sample variance, and low-frequency periodogram ordinates, designed so that, at the boundary case $(d=1/2)$, it admits a tractable limit distribution. These results are applied to construct a consistent parameter-free test of nonstationarity against long memory stationarity.