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ABSTRACT Harmonizable processes are a class of nonstationary time series, that are characterized by their dependence between different frequencies of a time series. The covariance between two frequencies is the dual frequency spectral density, an object analogous to the spectral density function. Local stationarity is another popular form of nonstationarity, though thus far, little attention has been paid to the dual frequency spectral density of a locally stationary process. The focus of this paper is on the dual frequency spectral density of local stationary time series and locally periodic stationary time series, its natural extension. We show that there are some subtle but important differences between the dual frequency spectral density of an almost periodic stationary process and a locally periodic stationary time series. Estimation of the dual frequency spectral density is typically done by smoothing the dual frequency periodogram. We study the sampling properties of this estimator under the assumption of locally periodic stationarity. In particular, we obtain a Gaussian approximation for the smoothed dual frequency periodogram over a group of frequencies, allowing for the number of frequency lags to grow with sample size. These results are used to test for correlation between different frequency bands in the time series. The variance of the smooth dual frequency periodogram is quite complex. However, by identifying which covariances are the most pertinent we propose a nonparametric method for consistently estimating the variance. This is necessary for constructing confidence intervals or testing aspects of the dual frequency spectral density. Simulations are given to illustrate our results. 

Graphical models are ubiquitous for summarizing conditional relations in multivariate data. In many applications involving multivariate time series, it is of interest to learn an interaction graph that treats each individual time series as nodes of the graph, with the presence of an edge between two nodes signifying conditional dependence given the others. Typically, the partial covariance is used as a measure of conditional dependence. However, in many applications, the outcomes may not be Gaussian and/or could be a mixture of different outcomes. For such time series using the partial covariance as a measure of conditional dependence may be restrictive. In this article, we propose a broad class of time series models which are specifically designed to succinctly encode process-wide conditional independence in its parameters. For each univariate component in the time series, we model its conditional distribution with a distribution from the exponential family. We develop a notion of process-wide compatibility under which such conditional specifications can be stitched together to form a well-defined strictly stationary multivariate time series. We call this construction a conditionally exponential stationary graphical model (CEStGM). A central quantity underlying CEStGM is a positive kernel which we call the interaction kernel. Spectral properties of such positive kernel operators constitute a core technical foundation of this work. We establish process-wide local and global Markov properties of CEStGM exploiting a Hammersley-Clifford type decomposition of the interaction kernel. Further, we study various probabilistic properties of CEStGM and show that it is geometrically mixing. An approximate Gibbs sampler is also developed to simulate sample paths of CEStGM. 

For multivariate stationary time series many important properties, such as partial correlation, graphical models and autoregressive representations are encoded in the inverse of its spectral density matrix. This is not true for nonstationary time series, where the pertinent information lies in the inverse infinite dimensional covariance matrix operator associated with the multivariate time series. This necessitates the study of the covariance of a multivariate nonstationary time series and its relationship to its inverse. We show that if the rows/columns of the infinite dimensional covariance matrix decay at a certain rate then the rate (up to a factor) transfers to the rows/columns of the inverse covariance matrix. This is used to obtain a nonstationary autoregressive representation of the time series and a Baxter-type bound between the parameters of the autoregressive infinite representation and the corresponding finite autoregressive projection. The aforementioned results lay the foundation for the subsequent analysis of locally stationary time series. In particular, we show that smoothness properties on the covariance matrix transfer to (i) the inverse covariance (ii) the parameters of the vector autoregressive representation and (iii) the partial covariances. All results are set up in such a way that the constants involved depend only on the eigenvalue of the covariance matrix and can be applied in the high-dimensional settings with non-diverging eigenvalues.