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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.

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 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.

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.

In working processes, mechanical systems are often affected by both internal and external forces, which are the cause of the forced vibrations of the structures. They can be destroyed if the amplitude of vibration reaches a high enough value. One of the most popular ways to reduce these forced vibrations is to attach tuned mass damper (TMD) devices, which are commonly added at the maximum displacement point of the structures. This paper presents the computed results of the free vibration and the vibration response of the space frame system under an external random load, which is described as a stationary process with white noise. Static and dynamic equations are formed through the finite element method. In addition, this work also establishes artificial neural networks (ANNs) in order to predict the vibration response of the first frequencies of the structure. Numerical studies show that the data set of the TMD device strongly affects the first frequencies of the mechanical system, and the proposed artificial intelligence (AI) model can predict exactly the vibration response of the first frequencies of the structure. For the forced vibration problem, we can find optimal parameters of the TMD device and thus obtain minimum displacements of the structure. The results of this work can be used as a reference when applying this type of structure to TMD devices.

In today’s world, smart electric vehicles are deeply integrated with smart energy, smart transportation and smart cities. In electric vehicles (EVs), owing to the harsh working conditions, mechanical parts are prone to fatigue damages, which endanger the driving safety of EVs. The practice has proved that the identification of periodic impact characteristics (PICs) can effectively indicate mechanical faults. This paper proposes a novel model-based approach for intelligent fault diagnosis of mechanical transmission train in EVs. The essential idea of this approach lies in the fusion of statistical information and model information from a dynamic process. In the algorithm, a novel fractal wavelet decomposition (FWD) is used to investigate the time-frequency representation of the input signal. Based on the sparsity of the PIC model in the Hilbert envelope spectrum, a method for evaluating PIC energy ratio (PICER) is defined based on an over-complete Fourier dictionary. A compound indicator considering kurtosis and PICER of dynamic signal is designed. Using this index, evaluations of the impulsiveness of the cycle-stationary process can be enabled, thus avoiding serious interference from the sporadic impact during measurements. The robustness of the proposed approach to noise is demonstrated via numerical simulations, and an engineering application is employed to validate its effectiveness.

We obtain a sharp convergence rate for banded covariance matrix estimates of stationary processes. A precise order of magnitude is derived for spectral radius of sample covariance matrices. We also consider a thresholded covariance matrix estimator that can better characterize sparsity if the true covariance matrix is sparse. As our main tool, we implement Toeplitz [Math. Ann. 70 (1911) 351–376] idea and relate eigenvalues of covariance matrices to the spectral densities or Fourier transforms of the variances. We develop a large deviation result for quadratic forms of stationary processes using m-dependence approximation, under the framework of causal representation and physical dependence measures.

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.

Numerous applications all the way from biology and physics to economics depend on the density of first crossings over a boundary. Motivated by the lack of general purpose analytical tools for computing first-passage time densities (FPTDs) for complex problems, we propose a new simple method based on the independent interval approximation (IIA). We generalise previous formulations of the IIA to include arbitrary initial conditions as well as to deal with discrete time and non-smooth continuous time processes. We derive a closed form expression for the FPTD in z and Laplace-transform space to a boundary in one dimension. Two classes of problems are analysed in detail: discrete time symmetric random walks (Markovian) and continuous time Gaussian stationary processes (Markovian and non-Markovian). Our results are in good agreement with Langevin dynamics simulations.