Zeitschriftenartikel: „Multivariate stationary process“

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MBEKE, Kévin Stanislas, and Ouagnina Hili. "Estimation of a stationary multivariate ARFIMA process." Afrika Statistika 13, no. 3 (2018): 1717–32. http://dx.doi.org/10.16929/as/1717.130.

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2

Cheng, R., and M. Pourahmadi. "The mixing rate of a stationary multivariate process." Journal of Theoretical Probability 6, no. 3 (1993): 603–17. http://dx.doi.org/10.1007/bf01066720.

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3

Latour, Alain. "The Multivariate Ginar(p) Process." Advances in Applied Probability 29, no. 1 (1997): 228–48. http://dx.doi.org/10.2307/1427868.

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A criterion is given for the existence of a stationary and causal multivariate integer-valued autoregressive process, MGINAR(p). The autocovariance function of this process being identical to the autocovariance function of a standard Gaussian MAR(p), we deduce that the MGINAR(p) process is nothing but a MAR(p) process. Consequently, the spectral density is directly found and gives good insight into the stochastic structure of a MGINAR(p). The estimation of parameters of the model, as well as the forecasting of the series, is discussed.

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Latour, Alain. "The Multivariate Ginar(p) Process." Advances in Applied Probability 29, no. 01 (1997): 228–48. http://dx.doi.org/10.1017/s0001867800027865.

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A criterion is given for the existence of a stationary and causal multivariate integer-valued autoregressive process, MGINAR(p). The autocovariance function of this process being identical to the autocovariance function of a standard Gaussian MAR(p), we deduce that the MGINAR(p) process is nothing but a MAR(p) process. Consequently, the spectral density is directly found and gives good insight into the stochastic structure of a MGINAR(p). The estimation of parameters of the model, as well as the forecasting of the series, is discussed.

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Sun, Ying, Ning Su, and Yue Wu. "Multivariate stationary non-Gaussian process simulation for wind pressure fields." Earthquake Engineering and Engineering Vibration 15, no. 4 (2016): 729–42. http://dx.doi.org/10.1007/s11803-016-0361-x.

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6

Borovkov, K., and G. Last. "On Rice's Formula for Stationary Multivariate Piecewise Smooth Processes." Journal of Applied Probability 49, no. 02 (2012): 351–63. http://dx.doi.org/10.1017/s002190020000913x.

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LetX= {Xt:t≥ 0} be a stationary piecewise continuousRd-valued process that moves between jumps along the integral curves of a given continuous vector field, and letS⊂Rdbe a smooth surface. The aim of this paper is to derive a multivariate version of Rice's formula, relating the intensity of the point process of (localized) continuous crossings ofSbyXto the distribution ofX0. Our result is illustrated by examples relating to queueing networks and stress release network models.

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Zhang, Zhengjun, and Richard L. Smith. "The behavior of multivariate maxima of moving maxima processes." Journal of Applied Probability 41, no. 4 (2004): 1113–23. http://dx.doi.org/10.1239/jap/1101840556.

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In the characterization of multivariate extremal indices of multivariate stationary processes, multivariate maxima of moving maxima processes, or M4 processes for short, have been introduced by Smith and Weissman. Central to the introduction of M4 processes is that the extreme observations of multivariate stationary processes may be characterized in terms of a limiting max-stable process under quite general conditions, and that a max-stable process can be arbitrarily closely approximated by an M4 process. In this paper, we derive some additional basic probabilistic properties for a finite class of M4 processes, each of which contains finite-range clustered moving patterns, called signature patterns, when extreme events occur. We use these properties to construct statistical estimation schemes for model parameters.

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Zhang, Zhengjun, and Richard L. Smith. "The behavior of multivariate maxima of moving maxima processes." Journal of Applied Probability 41, no. 04 (2004): 1113–23. http://dx.doi.org/10.1017/s0021900200020878.

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In the characterization of multivariate extremal indices of multivariate stationary processes, multivariate maxima of moving maxima processes, or M4 processes for short, have been introduced by Smith and Weissman. Central to the introduction of M4 processes is that the extreme observations of multivariate stationary processes may be characterized in terms of a limiting max-stable process under quite general conditions, and that a max-stable process can be arbitrarily closely approximated by an M4 process. In this paper, we derive some additional basic probabilistic properties for a finite class of M4 processes, each of which contains finite-range clustered moving patterns, called signature patterns, when extreme events occur. We use these properties to construct statistical estimation schemes for model parameters.

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Borovkov, K., and G. Last. "On Rice's Formula for Stationary Multivariate Piecewise Smooth Processes." Journal of Applied Probability 49, no. 2 (2012): 351–63. http://dx.doi.org/10.1239/jap/1339878791.

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Let X = {Xt: t ≥ 0} be a stationary piecewise continuous Rd-valued process that moves between jumps along the integral curves of a given continuous vector field, and let S ⊂ Rd be a smooth surface. The aim of this paper is to derive a multivariate version of Rice's formula, relating the intensity of the point process of (localized) continuous crossings of S by X to the distribution of X0. Our result is illustrated by examples relating to queueing networks and stress release network models.

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Gordy, Michael B. "Finite-Dimensional Distributions of a Square-Root Diffusion." Journal of Applied Probability 51, no. 4 (2014): 930–42. http://dx.doi.org/10.1239/jap/1421763319.

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We derive multivariate moment generating functions for the conditional and stationary distributions of a discrete sample path of n observations of a square-root diffusion (CIR) process, X(t). For any fixed vector of observation times t1,…,tn, we find the conditional joint distribution of (X(t1),…,X(tn)) is a multivariate noncentral chi-squared distribution and the stationary joint distribution is a Krishnamoorthy-Parthasarathy multivariate gamma distribution. Multivariate cumulants of the stationary distribution have a simple and computationally tractable expression. We also obtain the moment generating function for the increment X(t + δ) - X(t), and show that the increment is equivalent in distribution to a scaled difference of two independent draws from a gamma distribution.

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Gordy, Michael B. "Finite-Dimensional Distributions of a Square-Root Diffusion." Journal of Applied Probability 51, no. 04 (2014): 930–42. http://dx.doi.org/10.1017/s002190020001189x.

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We derive multivariate moment generating functions for the conditional and stationary distributions of a discrete sample path of n observations of a square-root diffusion (CIR) process, X(t). For any fixed vector of observation times t 1,…,t n , we find the conditional joint distribution of (X(t 1),…,X(t n )) is a multivariate noncentral chi-squared distribution and the stationary joint distribution is a Krishnamoorthy-Parthasarathy multivariate gamma distribution. Multivariate cumulants of the stationary distribution have a simple and computationally tractable expression. We also obtain the moment generating function for the increment X(t + δ) - X(t), and show that the increment is equivalent in distribution to a scaled difference of two independent draws from a gamma distribution.

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Mbeke, K. Stanislas, and Ouagnina Hili. "MINIMUM HELLINGER DISTANCE ESTIMATION OF A STATIONARY MULTIVARIATE LONG MEMORY ARFIMA PROCESS." Journal of Mathematical Sciences: Advances and Applications 50, no. 1 (2018): 13–36. http://dx.doi.org/10.18642/jmsaa_7100121930.

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13

BARONE, PIERO. "ON THE UNIVERSALITY OF THE DISTRIBUTION OF THE GENERALIZED EIGENVALUES OF A PENCIL OF HANKEL RANDOM MATRICES." Random Matrices: Theory and Applications 02, no. 01 (2013): 1250014. http://dx.doi.org/10.1142/s2010326312500141.

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Universality properties of the distribution of the generalized eigenvalues of a pencil of random Hankel matrices, arising in the solution of the exponential interpolation problem of a complex discrete stationary process, are proved under the assumption that every finite set of random variables of the process have a multivariate spherical distribution. An integral representation of the condensed density of the generalized eigenvalues is also derived. The asymptotic behavior of this function turns out to depend only on stationarity and not on the specific distribution of the process.

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Resnick, Sidney, and Rishin Roy. "Multivariate extremal processes, leader processes and dynamic choice models." Advances in Applied Probability 22, no. 2 (1990): 309–31. http://dx.doi.org/10.2307/1427538.

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Let (Y(t), t > 0) be a d-dimensional non-homogeneous multivariate extremal process. We suppose the ith component of Y describes time-dependent behaviour of random utilities associated with the ith choice. At time t we choose the ith alternative if the ith component of Y(t) is the largest of all the components. Let J(t) be the index of the largest component at time t so J has range {1, …, d} and call {J(t)} the leader process. Let Z(t) be the value of the largest component at time t. Then the bivariate process (J(t), Z(t)} is Markov. We discuss when J(t) and Z(t) are independent, when {J(s), 0<s≦t} and Z(t) are independent and when J(t) and {Z(s), 0<s≦t} are independent. In usual circumstances, {J(t)} is Markov and particular properties are given when the underlying distribution is max-stable. In the max-stable time-homogeneous case, {J(et)} is a stationary Markov chain with stationary transition probabilities.

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Resnick, Sidney, and Rishin Roy. "Multivariate extremal processes, leader processes and dynamic choice models." Advances in Applied Probability 22, no. 02 (1990): 309–31. http://dx.doi.org/10.1017/s0001867800019595.

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Let ( Y (t), t &gt; 0) be a d-dimensional non-homogeneous multivariate extremal process. We suppose the ith component of Y describes time-dependent behaviour of random utilities associated with the ith choice. At time t we choose the ith alternative if the ith component of Y (t) is the largest of all the components. Let J(t) be the index of the largest component at time t so J has range {1, …, d} and call {J(t)} the leader process. Let Z(t) be the value of the largest component at time t. Then the bivariate process (J(t), Z(t)} is Markov. We discuss when J(t) and Z(t) are independent, when {J(s), 0&lt;s≦t} and Z(t) are independent and when J(t) and {Z(s), 0&lt;s≦t} are independent. In usual circumstances, {J(t)} is Markov and particular properties are given when the underlying distribution is max-stable. In the max-stable time-homogeneous case, {J(et )} is a stationary Markov chain with stationary transition probabilities.

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BRÄNDÉN, P., M. LEANDER, and M. VISONTAI. "Multivariate Eulerian Polynomials and Exclusion Processes." Combinatorics, Probability and Computing 25, no. 4 (2016): 486–99. http://dx.doi.org/10.1017/s0963548316000031.

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We give a new combinatorial interpretation of the stationary distribution of the (partially) asymmetric exclusion process on a finite number of sites in terms of decorated alternative trees and coloured permutations. The corresponding expressions of the multivariate partition functions are then related to multivariate generalisations of Eulerian polynomials for coloured permutations considered recently by N. Williams and the third author, and others. We also discuss stability and negative dependence properties satisfied by the partition functions.

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Chung, Ching-Fan. "SAMPLE MEANS, SAMPLE AUTOCOVARIANCES, AND LINEAR REGRESSION OF STATIONARY MULTIVARIATE LONG MEMORY PROCESSES." Econometric Theory 18, no. 1 (2002): 51–78. http://dx.doi.org/10.1017/s0266466602181047.

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We develop an asymptotic theory for the first two sample moments of a stationary multivariate long memory process under fairly general conditions. In this theory the convergence rates and the limits (the fractional Brownian motion, the Rosenblatt process, etc.) all depend intrinsically on the degree of long memory in the process. The theory of the sample moments is then applied to the multiple linear regression model. An interesting finding is that, even though all the regressors and the disturbance are stationary and ergodic, the joint long memory in one single regressor and in the disturbance can invalidate the usual asymptotic theory for the ordinary least squares (OLS) estimation. Specifically, the convergence rates of the OLS estimators become slower, the limits are not normal, and the standard t- and F-tests all collapse.

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Bolla, Marianna, Tamás Szabados, Máté Baranyi, and Fatma Abdelkhalek. "Block circulant matrices and the spectra of multivariate stationary sequences." Special Matrices 9, no. 1 (2021): 36–51. http://dx.doi.org/10.1515/spma-2020-0121.

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Abstract Given a weakly stationary, multivariate time series with absolutely summable autocovariances, asymptotic relation is proved between the eigenvalues of the block Toeplitz matrix of the first n autocovariances and the union of spectra of the spectral density matrices at the n Fourier frequencies, as n → ∞. For the proof, eigenvalues and eigenvectors of block circulant matrices are used. The proved theorem has important consequences as for the analogies between the time and frequency domain calculations. In particular, the complex principal components are used for low-rank approximation of the process; whereas, the block Cholesky decomposition of the block Toeplitz matrix gives rise to dimension reduction within the innovation subspaces. The results are illustrated on a financial time series.

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19

Baccelli, François. "Stochastic ordering of random processes with an imbedded point process." Journal of Applied Probability 28, no. 3 (1991): 553–67. http://dx.doi.org/10.2307/3214491.

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We introduce multivariate partial orderings related with the Palm and time-stationary probabilities of a point process. Using these orderings, we give conditions for the monotonicity of a random sequence, with respect to some integral stochastic ordering, to be inherited with a continuous time process in which this sequence is imbedded. This type of inheritance is also discussed for the property of association.

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Baccelli, François. "Stochastic ordering of random processes with an imbedded point process." Journal of Applied Probability 28, no. 03 (1991): 553–67. http://dx.doi.org/10.1017/s0021900200042418.

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We introduce multivariate partial orderings related with the Palm and time-stationary probabilities of a point process. Using these orderings, we give conditions for the monotonicity of a random sequence, with respect to some integral stochastic ordering, to be inherited with a continuous time process in which this sequence is imbedded. This type of inheritance is also discussed for the property of association.

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Raath, Kim C., Katherine B. Ensor, Alena Crivello, and David W. Scott. "Denoising Non-Stationary Signals via Dynamic Multivariate Complex Wavelet Thresholding." Entropy 25, no. 11 (2023): 1546. http://dx.doi.org/10.3390/e25111546.

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Over the past few years, we have seen an increased need to analyze the dynamically changing behaviors of economic and financial time series. These needs have led to significant demand for methods that denoise non-stationary time series across time and for specific investment horizons (scales) and localized windows (blocks) of time. Wavelets have long been known to decompose non-stationary time series into their different components or scale pieces. Recent methods satisfying this demand first decompose the non-stationary time series using wavelet techniques and then apply a thresholding method to separate and capture the signal and noise components of the series. Traditionally, wavelet thresholding methods rely on the discrete wavelet transform (DWT), which is a static thresholding technique that may not capture the time series of the estimated variance in the additive noise process. We introduce a novel continuous wavelet transform (CWT) dynamically optimized multivariate thresholding method (WaveL2E). Applying this method, we are simultaneously able to separate and capture the signal and noise components while estimating the dynamic noise variance. Our method shows improved results when compared to well-known methods, especially for high-frequency signal-rich time series, typically observed in finance.

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22

Campi, Marco. "A unique representation theorem for the conditional expectation of stationary processes and application to dynamic estimation problems." Journal of Applied Probability 34, no. 2 (1997): 372–80. http://dx.doi.org/10.2307/3215377.

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In this paper, multivariate strict sense stationary stochastic processes are considered. It is shown that there exists a universal function by means of which the conditional expectation of any stationary process with respect to its past can be represented. This requires no ergodicity assumptions. The important implications of this result in the evaluation of the achievable performance in certain dynamic estimation problems with incomplete statistical information are also discussed.

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Campi, Marco. "A unique representation theorem for the conditional expectation of stationary processes and application to dynamic estimation problems." Journal of Applied Probability 34, no. 02 (1997): 372–80. http://dx.doi.org/10.1017/s0021900200101019.

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In this paper, multivariate strict sense stationary stochastic processes are considered. It is shown that there exists a universal function by means of which the conditional expectation of any stationary process with respect to its past can be represented. This requires no ergodicity assumptions. The important implications of this result in the evaluation of the achievable performance in certain dynamic estimation problems with incomplete statistical information are also discussed.

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Passeggeri, Riccardo, and Almut E. D. Veraart. "Limit theorems for multivariate Brownian semistationary processes and feasible results." Advances in Applied Probability 51, no. 03 (2019): 667–716. http://dx.doi.org/10.1017/apr.2019.30.

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AbstractIn this paper we introduce the multivariate Brownian semistationary (BSS) process and study the joint asymptotic behaviour of its realised covariation using in-fill asymptotics. First, we present a central limit theorem for general multivariate Gaussian processes with stationary increments, which are not necessarily semimartingales. Then, we show weak laws of large numbers, central limit theorems, and feasible results for BSS processes. An explicit example based on the so-called gamma kernels is also provided.

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Klein, André, and Guy Mélard. "An Algorithm for the Fisher Information Matrix of a VARMAX Process." Algorithms 16, no. 8 (2023): 364. http://dx.doi.org/10.3390/a16080364.

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In this paper, an algorithm for Mathematica is proposed for the computation of the asymptotic Fisher information matrix for a multivariate time series, more precisely for a controlled vector autoregressive moving average stationary process, or VARMAX process. Meanwhile, we present briefly several algorithms published in the literature and discuss the sufficient condition of invertibility of that matrix based on the eigenvalues of the process operators. The results are illustrated by numerical computations.

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Kim, Tae-Sung, Mi-Hwa Ko, and Sung-Mo Chung. "A CENTRAL LIMIT THEOREM FOR THE STATIONARY MULTIVARIATE LINEAR PROCESS GENERATED BY ASSOCIATED RANDOM VICTORS." Communications of the Korean Mathematical Society 17, no. 1 (2002): 95–102. http://dx.doi.org/10.4134/ckms.2002.17.1.095.

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27

Perrin, Olivier, and Martin Schlather. "Can any multivariate gaussian vector be interpreted as a sample from a stationary random process?" Statistics & Probability Letters 77, no. 9 (2007): 881–84. http://dx.doi.org/10.1016/j.spl.2006.12.005.

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Entezami, Alireza, and Hashem Shariatmadar. "Damage localization under ambient excitations and non-stationary vibration signals by a new hybrid algorithm for feature extraction and multivariate distance correlation methods." Structural Health Monitoring 18, no. 2 (2018): 347–75. http://dx.doi.org/10.1177/1475921718754372.

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Ambient excitations applied to structures may lead to non-stationary vibration responses. In such circumstances, it may be difficult or improper to extract meaningful and significant damage features through methods that mainly rely on the stationarity of data. This article proposes a new hybrid algorithm for feature extraction as a combination of a new adaptive signal decomposition method called improved complete ensemble empirical mode decomposition with adaptive noise and autoregressive moving average model. The major contribution of this algorithm is to address the important issue of feature extraction under ambient vibration and non-stationary signals. The improved complete ensemble empirical mode decomposition with adaptive noise method is an improvement on the well-known ensemble empirical mode decomposition technique by removing redundant intrinsic mode functions. In addition, a novel automatic approach is presented to select the most relevant intrinsic mode functions to damage based on the intrinsic mode function energy level. Fitting an autoregressive moving average model to each selected intrinsic mode function, the model residuals are extracted as the damage-sensitive features. The main limitation is that such features are high-dimensional multivariate time series data, which may make a difficult and time-consuming decision-making process for damage localization. Multivariate distance correlation methods are introduced to cope with this drawback and locate structural damage using the multivariate residual sets of the normal and damaged conditions. The accuracy and robustness of the proposed methods are validated by a numerical shear-building model and an experimental benchmark structure. The effects of sampling frequency and time duration are evaluated as well. Results demonstrate the effectiveness and capability of the proposed methods to extract sufficient and reliable features, identify damage location, and quantify damage severity under ambient excitations and non-stationary signals.

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Last, Günter, Mathew D. Penrose, Matthias Schulte, and Christoph Thäle. "Moments and Central Limit Theorems for Some Multivariate Poisson Functionals." Advances in Applied Probability 46, no. 2 (2014): 348–64. http://dx.doi.org/10.1239/aap/1401369698.

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This paper deals with Poisson processes on an arbitrary measurable space. Using a direct approach, we derive formulae for moments and cumulants of a vector of multiple Wiener-Itô integrals with respect to the compensated Poisson process. Also, we present a multivariate central limit theorem for a vector whose components admit a finite chaos expansion of the type of a Poisson U-statistic. The approach is based on recent results of Peccati et al. (2010), combining Malliavin calculus and Stein's method; it also yields Berry-Esseen-type bounds. As applications, we discuss moment formulae and central limit theorems for general geometric functionals of intersection processes associated with a stationary Poisson process of k-dimensional flats in .

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Last, Günter, Mathew D. Penrose, Matthias Schulte, and Christoph Thäle. "Moments and Central Limit Theorems for Some Multivariate Poisson Functionals." Advances in Applied Probability 46, no. 02 (2014): 348–64. http://dx.doi.org/10.1017/s0001867800007126.

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This paper deals with Poisson processes on an arbitrary measurable space. Using a direct approach, we derive formulae for moments and cumulants of a vector of multiple Wiener-Itô integrals with respect to the compensated Poisson process. Also, we present a multivariate central limit theorem for a vector whose components admit a finite chaos expansion of the type of a PoissonU-statistic. The approach is based on recent results of Peccatiet al.(2010), combining Malliavin calculus and Stein's method; it also yields Berry-Esseen-type bounds. As applications, we discuss moment formulae and central limit theorems for general geometric functionals of intersection processes associated with a stationary Poisson process ofk-dimensional flats in.

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Su, Yan Wen, Guo Qing Huang, and Liu Liu Peng. "Time-Frequency Analysis of Non-Stationary Ground Motions via Multivariate Empirical Mode Decomposition." Applied Mechanics and Materials 580-583 (July 2014): 1734–41. http://dx.doi.org/10.4028/www.scientific.net/amm.580-583.1734.

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In this paper, MEMD-based scalogram and coscalogram, and instantaneous frequency spectral are proposed to characterize the data derived from the multivariate non-stationary process. The scalogram and instantaneous frequency spectral capture spectral evolution of each component while the coscalogram reveals embedded intermittent correlation between two components. Compared with scale-based scalogram and coscalogram, frequency-based instantaneous frequency spectral provides more detailed portrayal for multivariate data. The effectiveness of the proposed MEMD-based time-frequency analysis framework is validated by numerical examples of uniformly and generally modulated ground motions.

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Sundararajan, Raanju R., Ron Frostig, and Hernando Ombao. "Modeling Spectral Properties in Stationary Processes of Varying Dimensions with Applications to Brain Local Field Potential Signals." Entropy 22, no. 12 (2020): 1375. http://dx.doi.org/10.3390/e22121375.

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In some applications, it is important to compare the stochastic properties of two multivariate time series that have unequal dimensions. A new method is proposed to compare the spread of spectral information in two multivariate stationary processes with different dimensions. To measure discrepancies, a frequency specific spectral ratio (FS-ratio) statistic is proposed and its asymptotic properties are derived. The FS-ratio is blind to the dimension of the stationary process and captures the proportion of spectral power in various frequency bands. Here we develop a technique to automatically identify frequency bands that carry significant spectral power. We apply our method to track changes in the complexity of a 32-channel local field potential (LFP) signal from a rat following an experimentally induced stroke. At every epoch (a distinct time segment from the duration of the experiment), the nonstationary LFP signal is decomposed into stationary and nonstationary latent sources and the complexity is analyzed through these latent stationary sources and their dimensions that can change across epochs. The analysis indicates that spectral information in the Beta frequency band (12–30 Hertz) demonstrated the greatest change in structure and complexity due to the stroke.

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Segers, Johan. "Functionals of clusters of extremes." Advances in Applied Probability 35, no. 4 (2003): 1028–45. http://dx.doi.org/10.1239/aap/1067436333.

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For arbitrary stationary sequences of random variables satisfying a mild mixing condition, distributional approximations are established for functionals of clusters of exceedances over a high threshold. The approximations are in terms of the distribution of the process conditionally on the event that the first variable exceeds the threshold. This conditional distribution is shown to converge to a nontrivial limit if the finite-dimensional distributions of the process are in the domain of attraction of a multivariate extreme-value distribution. In this case, therefore, limit distributions are obtained for functionals of clusters of extremes, thereby generalizing results for higher-order stationary Markov chains by Yun (2000).

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Segers, Johan. "Functionals of clusters of extremes." Advances in Applied Probability 35, no. 04 (2003): 1028–45. http://dx.doi.org/10.1017/s0001867800012726.

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For arbitrary stationary sequences of random variables satisfying a mild mixing condition, distributional approximations are established for functionals of clusters of exceedances over a high threshold. The approximations are in terms of the distribution of the process conditionally on the event that the first variable exceeds the threshold. This conditional distribution is shown to converge to a nontrivial limit if the finite-dimensional distributions of the process are in the domain of attraction of a multivariate extreme-value distribution. In this case, therefore, limit distributions are obtained for functionals of clusters of extremes, thereby generalizing results for higher-order stationary Markov chains by Yun (2000).

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35

Peng, Bo, Jun Xu, and Yongbo Peng. "Efficient simulation of multivariate non-stationary ground motions based on a virtual continuous process and EOLE." Mechanical Systems and Signal Processing 184 (February 2023): 109722. http://dx.doi.org/10.1016/j.ymssp.2022.109722.

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36

Al-Awadhi, F. A. "A multivariate prediction of spatial process with non-stationary covariance for Kuwait non-methane hydrocarbons levels." Environmental and Ecological Statistics 18, no. 1 (2009): 57–77. http://dx.doi.org/10.1007/s10651-009-0120-5.

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37

Moser, Martin, and Robert Stelzer. "Tail behavior of multivariate lévy-driven mixed moving average processes and supOU Stochastic Volatility Models." Advances in Applied Probability 43, no. 4 (2011): 1109–35. http://dx.doi.org/10.1239/aap/1324045701.

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Multivariate Lévy-driven mixed moving average (MMA) processes of the type Xt = ∬f(A, t - s)Λ(dA, ds) cover a wide range of well known and extensively used processes such as Ornstein-Uhlenbeck processes, superpositions of Ornstein-Uhlenbeck (supOU) processes, (fractionally integrated) continuous-time autoregressive moving average processes, and increments of fractional Lévy processes. In this paper we introduce multivariate MMA processes and give conditions for their existence and regular variation of the stationary distributions. Furthermore, we study the tail behavior of multivariate supOU processes and of a stochastic volatility model, where a positive semidefinite supOU process models the stochastic volatility.

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Moser, Martin, and Robert Stelzer. "Tail behavior of multivariate lévy-driven mixed moving average processes and supOU Stochastic Volatility Models." Advances in Applied Probability 43, no. 04 (2011): 1109–35. http://dx.doi.org/10.1017/s0001867800005322.

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Multivariate Lévy-driven mixed moving average (MMA) processes of the type X t = ∬f(A, t - s)Λ(dA, ds) cover a wide range of well known and extensively used processes such as Ornstein-Uhlenbeck processes, superpositions of Ornstein-Uhlenbeck (supOU) processes, (fractionally integrated) continuous-time autoregressive moving average processes, and increments of fractional Lévy processes. In this paper we introduce multivariate MMA processes and give conditions for their existence and regular variation of the stationary distributions. Furthermore, we study the tail behavior of multivariate supOU processes and of a stochastic volatility model, where a positive semidefinite supOU process models the stochastic volatility.

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39

Illsley, Robert. "The moments of the number of exits from a simply connected region." Advances in Applied Probability 30, no. 1 (1998): 167–80. http://dx.doi.org/10.1239/aap/1035227998.

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We generalise the work of Cramér and Leadbetter, Ylvisaker and Ito on the level crossings of a stationary Gaussian process to multivariate processes. Necessary and sufficient conditions for the existence of the expected number of crossings E(C) of the boundary of a region of ℝp by a stationary vector stochastic process are obtained, along with an explicit formula for E(C) in the Gaussian case. A rigorous proof of Belyaev's integral formula for the factorial moments of the number of exits from a region of ℝp is given for a class of processes which includes Gaussian processes having a finite second order spectral moment matrix. Applications to χ2 processes are briefly considered.

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40

Illsley, Robert. "The moments of the number of exits from a simply connected region." Advances in Applied Probability 30, no. 01 (1998): 167–80. http://dx.doi.org/10.1017/s0001867800008144.

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We generalise the work of Cramér and Leadbetter, Ylvisaker and Ito on the level crossings of a stationary Gaussian process to multivariate processes. Necessary and sufficient conditions for the existence of the expected number of crossings E(C) of the boundary of a region of ℝ p by a stationary vector stochastic process are obtained, along with an explicit formula for E(C) in the Gaussian case. A rigorous proof of Belyaev's integral formula for the factorial moments of the number of exits from a region of ℝ p is given for a class of processes which includes Gaussian processes having a finite second order spectral moment matrix. Applications to χ2 processes are briefly considered.

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41

Hult, Henrik, and Filip Lindskog. "On regular variation for infinitely divisible random vectors and additive processes." Advances in Applied Probability 38, no. 1 (2006): 134–48. http://dx.doi.org/10.1239/aap/1143936144.

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We study the tail behavior of regularly varying infinitely divisible random vectors and additive processes, i.e. stochastic processes with independent but not necessarily stationary increments. We show that the distribution of an infinitely divisible random vector is tail equivalent to its Lévy measure and we study the asymptotic decay of the probability for an additive process to hit sets far away from the origin. The results are extensions of known univariate results to the multivariate setting; we exemplify some of the difficulties that arise in the multivariate case.

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42

Hult, Henrik, and Filip Lindskog. "On regular variation for infinitely divisible random vectors and additive processes." Advances in Applied Probability 38, no. 01 (2006): 134–48. http://dx.doi.org/10.1017/s0001867800000847.

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We study the tail behavior of regularly varying infinitely divisible random vectors and additive processes, i.e. stochastic processes with independent but not necessarily stationary increments. We show that the distribution of an infinitely divisible random vector is tail equivalent to its Lévy measure and we study the asymptotic decay of the probability for an additive process to hit sets far away from the origin. The results are extensions of known univariate results to the multivariate setting; we exemplify some of the difficulties that arise in the multivariate case.

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43

Ball, Frank. "Central limit theorems for multivariate semi-Markov sequences and processes, with applications." Journal of Applied Probability 36, no. 2 (1999): 415–32. http://dx.doi.org/10.1239/jap/1032374462.

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In this paper, central limit theorems for multivariate semi-Markov sequences and processes are obtained, both as the number of jumps of the associated Markov chain tends to infinity and, if appropriate, as the time for which the process has been running tends to infinity. The theorems are widely applicable since many functions defined on Markov or semi-Markov processes can be analysed by exploiting appropriate embedded multivariate semi-Markov sequences. An application to a problem in ion channel modelling is described in detail. Other applications, including to multivariate stationary reward processes, counting processes associated with Markov renewal processes, the interpretation of Markov chain Monte Carlo runs and statistical inference on semi-Markov models are briefly outlined.

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44

Ball, Frank. "Central limit theorems for multivariate semi-Markov sequences and processes, with applications." Journal of Applied Probability 36, no. 02 (1999): 415–32. http://dx.doi.org/10.1017/s0021900200017228.

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In this paper, central limit theorems for multivariate semi-Markov sequences and processes are obtained, both as the number of jumps of the associated Markov chain tends to infinity and, if appropriate, as the time for which the process has been running tends to infinity. The theorems are widely applicable since many functions defined on Markov or semi-Markov processes can be analysed by exploiting appropriate embedded multivariate semi-Markov sequences. An application to a problem in ion channel modelling is described in detail. Other applications, including to multivariate stationary reward processes, counting processes associated with Markov renewal processes, the interpretation of Markov chain Monte Carlo runs and statistical inference on semi-Markov models are briefly outlined.

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45

Brachner, Claudia, Vicky Fasen, and Alexander Lindner. "Extremes of autoregressive threshold processes." Advances in Applied Probability 41, no. 2 (2009): 428–51. http://dx.doi.org/10.1239/aap/1246886618.

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In this paper we study the tail and the extremal behaviors of stationary solutions of threshold autoregressive (TAR) models. It is shown that a regularly varying noise sequence leads in general to only an O-regularly varying tail of the stationary solution. Under further conditions on the partition, it is shown however that TAR(S,1) models of order 1 with S regimes have regularly varying tails, provided that the noise sequence is regularly varying. In these cases, the finite-dimensional distribution of the stationary solution is even multivariate regularly varying and its extremal behavior is studied via point process convergence. In particular, a TAR model with regularly varying noise can exhibit extremal clusters. This is in contrast to TAR models with noise in the maximum domain of attraction of the Gumbel distribution and which is either subexponential or in ℒ(γ) with γ > 0. In this case it turns out that the tail of the stationary solution behaves like a constant times that of the noise sequence, regardless of the order and the specific partition of the TAR model, and that the process cannot exhibit clusters on high levels.

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46

Brachner, Claudia, Vicky Fasen, and Alexander Lindner. "Extremes of autoregressive threshold processes." Advances in Applied Probability 41, no. 02 (2009): 428–51. http://dx.doi.org/10.1017/s0001867800003360.

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In this paper we study the tail and the extremal behaviors of stationary solutions of threshold autoregressive (TAR) models. It is shown that a regularly varying noise sequence leads in general to only an O-regularly varying tail of the stationary solution. Under further conditions on the partition, it is shown however that TAR(S,1) models of order 1 with S regimes have regularly varying tails, provided that the noise sequence is regularly varying. In these cases, the finite-dimensional distribution of the stationary solution is even multivariate regularly varying and its extremal behavior is studied via point process convergence. In particular, a TAR model with regularly varying noise can exhibit extremal clusters. This is in contrast to TAR models with noise in the maximum domain of attraction of the Gumbel distribution and which is either subexponential or in ℒ(γ) with γ &gt; 0. In this case it turns out that the tail of the stationary solution behaves like a constant times that of the noise sequence, regardless of the order and the specific partition of the TAR model, and that the process cannot exhibit clusters on high levels.

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Góngora, Leonardo, Alessia Paglialonga, Alfonso Mastropietro, Giovanna Rizzo, and Riccardo Barbieri. "A Novel Approach for Segment-Length Selection Based on Stationarity to Perform Effective Connectivity Analysis Applied to Resting-State EEG Signals." Sensors 22, no. 13 (2022): 4747. http://dx.doi.org/10.3390/s22134747.

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Connectivity among different areas within the brain is a topic that has been notably studied in the last decade. In particular, EEG-derived measures of effective connectivity examine the directionalities and the exerted influences raised from the interactions among neural sources that are masked out on EEG signals. This is usually performed by fitting multivariate autoregressive models that rely on the stationarity that is assumed to be maintained over shorter bits of the signals. However, despite being a central condition, the selection process of a segment length that guarantees stationary conditions has not been systematically addressed within the effective connectivity framework, and thus, plenty of works consider different window sizes and provide a diversity of connectivity results. In this study, a segment-size-selection procedure based on fourth-order statistics is proposed to make an informed decision on the appropriate window size that guarantees stationarity both in temporal and spatial terms. Specifically, kurtosis is estimated as a function of the window size and used to measure stationarity. A search algorithm is implemented to find the segments with similar stationary properties while maximizing the number of channels that exhibit the same properties and grouping them accordingly. This approach is tested on EEG signals recorded from six healthy subjects during resting-state conditions, and the results obtained from the proposed method are compared to those obtained using the classical approach for mapping effective connectivity. The results show that the proposed method highlights the influence that arises in the Default Mode Network circuit by selecting a window of 4 s, which provides, overall, the most uniform stationary properties across channels.

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Barone, Piero. "A METHOD FOR GENERATING INDEPENDENT REALIZATIONS OF A MULTIVARIATE NORMAL STATIONARY AND INVERTIBLE ARMA(p, q) PROCESS." Journal of Time Series Analysis 8, no. 2 (1987): 125–30. http://dx.doi.org/10.1111/j.1467-9892.1987.tb00426.x.

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49

Lieberman, Offer, Judith Rousseau, and David M. Zucker. "VALID EDGEWORTH EXPANSION FOR THE SAMPLE AUTOCORRELATION FUNCTION UNDER LONG RANGE DEPENDENCE." Econometric Theory 17, no. 1 (2001): 257–75. http://dx.doi.org/10.1017/s0266466601171094.

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We prove in this paper the validity of an Edgeworth expansion to the joint distribution of the sample autocorrelations of a stationary Gaussian long memory process. The method of proof relies on a verification of the suitably modified conditions for the validity of a multivariate Edgeworth expansion of Durbin (1980, Biometrika 67, 311–333). A simulation study proves the expansion to be useful and accurate.

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Kella, Offer, and Wolfgang Stadje. "Markov-modulated linear fluid networks with Markov additive input." Journal of Applied Probability 39, no. 2 (2002): 413–20. http://dx.doi.org/10.1239/jap/1025131438.

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We consider a network of dams to which the external input is a multivariate Markov additive process. For each state of the Markov chain modulating the Markov additive process, the release rates are linear (constant multiple of the content level). Each unit of material processed by a given station is then divided into fixed proportions each of which is routed to another station or leaves the system. For each state of the modulating process, this routeing is determined by some substochastic matrix. We identify simple conditions for stability and show how to compute transient and stationary characteristics of such networks.

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