In the general setting of long-memory multivariate time series, the long-memory characteristics are defined by two components. The long-memory parameters describe the autocorrelation of each time series. And the long-run covariance measures the coupling between time series, with general phase parameters. It is of interest to estimate the long-memory, long-run covariance and general phase parameters of time series generated by this wide class of models although they are not necessarily Gaussian nor stationary. This estimation is thus not directly possible using real wavelets decomposition or Fourier analysis. Our purpose is to define an inference approach based on a representation using quasi-analytic wavelets. We first show that the covariance of the wavelet coefficients provides an adequate estimator of the covariance structure including the phase term. Consistent estimators based on a local Whittle approximation are then proposed. Simulations highlight a satisfactory behavior of the estimation on finite samples on linear time series and on multivariate fractional Brownian motions. An application on a real neuroscience dataset is presented, where long-memory and brain connectivity are inferred.

翻译：在长期记忆多元时间序列的一般设定中，长期记忆特征由两个分量定义：长期记忆参数描述每个时间序列的自相关性，而长期协方差则衡量时间序列之间的耦合性，并包含一般相位参数。尽管这些由广泛模型类生成的时间序列未必服从高斯分布或平稳性，但估计其长期记忆参数、长期协方差及一般相位参数具有重要意义。因此，直接使用实小波分解或傅里叶分析无法实现该估计。本文旨在定义一种基于拟解析小波表示的推断方法。我们首先证明，小波系数的协方差能够有效估计包含相位项的协方差结构。进而提出基于局部Whittle逼近的一致性估计量。模拟实验表明，该方法在有限样本的线性时间序列及多元分数布朗运动上具有良好表现。最后，我们将其应用于真实神经科学数据集，推断长期记忆特性与大脑连通性。