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Covariances and spectral density functions play a fundamental role in the theory of time series. There is a well-developed asymptotic theory for their estimates for low-dimensional stationary processes. For high-dimensional non-stationary processes, however, many important problems on their asymptotic behaviors are still unanswered. This paper presents a systematic asymptotic theory for the estimates of time-varying second-order statistics for a general class of high-dimensional locally stationary processes. Using the framework of functional dependence measure, we derive convergence rates of the estimates which depend on the sample size T, the dimension p, the moment condition and the dependence of the underlying processes.

We propose an adaptive bandwidth selector via cross validation for local M-estimators in locally stationary processes. We prove asymptotic optimality of the procedure under mild conditions on the underlying parameter curves. The results are applicable to a wide range of locally stationary processes such linear and nonlinear processes. A simulation study shows that the method works fairly well also in misspecified situations.

In this paper, we investigate the distributed convex optimization problem for a class of nonlinear multi-agent systems disturbed by random noise over a directed graph. The target problem involves designing a continuous-time algorithm to minimize the sum of all local cost functions associated with each agent. The target noise is considered as a second-order stationary process under mild assumptions. The noise-to-state exponential stability for the multi-agent system based on random differential equations is analyzed using a random field method. Sufficient conditions corresponding to the second moment relative to the optimal solution in the form of matrix inequalities are established. Then, the grid search method is employed to determine the best system parameters such that the second moment of the estimation error has the minimum value. In addition, the obtained results are applied to solve the average consensus problem in the presence of a stationary process. Finally, a numerical example is presented to verify the effectiveness of the proposed algorithm.

We adapt the classical definition of locally stationary processes in discrete time (see e.g. Dahlhaus, ‘Locally stationary processes’, in Time Series Analysis: Methods and Applications (2012)) to the continuous-time setting and obtain equivalent representations in the time and frequency domains. From this, a unique time-varying spectral density is derived using the Wigner–Ville spectrum. As an example, we investigate time-varying Lévy-driven state space processes, including the class of time-varying Lévy-driven CARMA processes. First, the connection between these two classes of processes is examined. Considering a sequence of time-varying Lévy-driven state space processes, we then give sufficient conditions on the coefficient functions that ensure local stationarity with respect to the given definition.

Periodicity and trend are features describing an observed sequence, and extracting these features is an important issue in many scientific fields. However, it is not an easy task for existing methods to analyse simultaneously the trend and dynamics of the periodicity such as time varying frequency and amplitude, and the adaptivity of the analysis to such dynamics and robustness to heteroscedastic dependent errors are not guaranteed. These tasks become even more challenging when there are multiple periodic components. We propose a non‐parametric model to describe the dynamics of multicomponent periodicity and investigate the recently developed synchro‐squeezing transform in extracting these features in the presence of a trend and heteroscedastic dependent errors. The identifiability problem of the non‐parametric periodicity model is studied, and the adaptivity and robustness properties of the synchro‐squeezing transform are theoretically justified in both discrete and continuous time settings. Consequently we have a new technique for decoupling the trend, periodicity and heteroscedastic, dependent error process in a general non‐parametric set‐up. Results of a series of simulations are provided, and the incidence time series of varicella and herpes zoster in Taiwan and respiratory signals observed from a sleep study are analysed.

Edgeworth–Cornish–Fisher expansions are hugely important, as they give the distribution, density and quantiles of any standard estimate. Here we show that the sample mean of a univariate or multivariate stationary process is a standard estimate, so that all the known results for standard estimates can be applied. We also show how to allow for missing data and weighted means.

This paper gives expansions for the distribution, density and quantiles of any estimate that is a smooth function of the sample cross-moments of a stationary process. Three versions of these are given, depending on whether an exact, approximate, or asymptotic form is used for the variance or covariance of the estimate. Eight examples are provided, including sample autocovariances and autocorrelations. Their Central Limit Theorems extend those in the literature, such as Bartlett’s formula, by allowing for the effect of the mean and higher order cross-cumulants. Their distribution and quantiles are given to magnitude n−r/2 up to r=3, where n is the sample size.

We study the prediction problem for deterministic stationary processes X(t) possessing spectral density f . We describe the asymptotic behavior of the best linear mean squared prediction error σn2(f) in predicting X(0) given X(t) , -n≤t≤-1 , as n goes to infinity. We consider a class of spectral densities of the form f=fdg , where fd is the spectral density of a deterministic process that has a very high order contact with zero due to which the Szegő condition is violated, while g is a nonnegative function that can have arbitrary power type singularities. We show that for spectral densities f from this class the prediction error σn2(f) behaves like a power as n→∞ . Examples illustrate the obtained results.

The additive model and the varying-coefficient model are both powerful regression tools, with wide practical applications. However, our empirical study on a financial data has shown that both of these models have drawbacks when applied to locally stationary time series. For the analysis of functional data, Zhang and Wang have proposed a flexible regression method, called the varying-coefficient additive model (VCAM), and presented a two-step spline estimation method. Motivated by their approach, we adopt the VCAM to characterize the time-varying regression function in a locally stationary context. We propose a three-step spline estimation method and show its consistency and asymptotic normality. For the purpose of model diagnosis, we suggest an L 2 -distance test statistic to check multiplicative assumption, and raise a two-stage penalty procedure to identify the additive terms and the varying-coefficient terms provided that the VCAM is applicable. We also present the asymptotic distribution of the proposed test statistics and demonstrate the consistency of the two-stage model identification procedure. Simulation studies investigating the finite-sample performance of the estimation and model diagnosis methods confirm the validity of our asymptotic theory. The financial data are also considered. Supplementary materials for this article are available online.

We study the variance of the number of zeroes of a stationary Gaussian process on a long interval. We give a simple asymptotic description under mild mixing conditions. This allows us to characterise minimal and maximal growth. We show that a small (symmetrised) atom in the spectral measure at a special frequency does not affect the asymptotic growth of the variance, while an atom at any other frequency results in maximal growth.