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.

翻译：我们研究具有长记忆性和非平稳性的时间序列中不同统计量的渐近行为。对于记忆参数$d\in(-1/2,3/2)$的过程，我们推导了在固定数量傅里叶频率下离散傅里叶变换的联合极限分布，并采用统一归一化处理。所得极限为高斯分布，具有显式的协方差结构。特别关注边界情况$d=1/2$，即所谓的$1/f$噪声。我们证明对数修正可使样本均值和样本方差产生非退化极限，从而得到$\chi^2$类型的显式渐近分布。我们构造了一个结合样本均值、样本方差和低频周期图坐标的统计量，该统计量在边界情况$(d=1/2)$下具有易于处理的极限分布。这些结果被应用于构造一个针对长记忆平稳性检验的非平稳性一致性无参数检验方法。