Asymptotic theory for stationary processes

Wu, Wei Biao

doi:10.4310/sii.2011.v4.n2.a15

Cited by 105 publications

(88 citation statements)

References 180 publications

(131 reference statements)

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“…See Borkar (1993), Tong (1990), Kallianpur (1981), Ornstein (1973) and Rosenblatt (2009) for more historical backgrounds on the above stochastic realization theory. See also Wu (2011) for examples of stationary processes that are of form (5). Following Priestley (1988) and Wu (2005), we can view (X i ) as a physical system with (ε j , ε j−1 , .…”

Section: Asymptotics Of Sample Auto-covariancesmentioning

confidence: 99%

Covariance Matrix Estimation in Time Series

Xiao

2012

Time Series Analysis: Methods and Applications

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Covariances play a fundamental role in the theory of time series and they are critical quantities that are needed in both spectral and time domain analysis. Estimation of covariance matrices is needed in the construction of confidence regions for unknown parameters, hypothesis testing, principal component analysis, prediction, discriminant analysis among others. In this paper we consider both low-and high-dimensional covariance matrix estimation problems and present a review for asymptotic properties of sample covariances and covariance matrix estimates. In particular, we shall provide an asymptotic theory for estimates of high dimensional covariance matrices in time series, and a consistency result for covariance matrix estimates for estimated parameters.

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Section: Asymptotics Of Sample Auto-covariancesmentioning

confidence: 99%

Covariance Matrix Estimation in Time Series

Xiao

2012

Time Series Analysis: Methods and Applications

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“…The following measure of dependence has been introduced by Wu [33] (see also [34]). Let ε 0 be a copy of ε 0 independent of (ε j ) j∈Z d and define…”

Section: Measure Of Dependencementioning

confidence: 99%

“…Several limit theorems are proved under p-stability, see [34] and references therein. An invariance principle for the set-indexed process (1) with b n,j (A) = λ(nA ∩ R j ) is obtained in [14].…”

Section: Introductionmentioning

confidence: 99%

Invariance principles for self-similar set-indexed random fields

Biermé

Durieu

2014

Trans. Amer. Math. Soc.

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Abstract. For a stationary random field (X j ) j∈Z d and some measure µ on R d , we consider the set-indexed weighted sum process Sn(A) = j∈Z d µ(nA ∩ R j ) 1 2 X j , where R j is the unit cube with lower corner j. We establish a general invariance principle under a p-stability assumption on the X j 's and an entropy condition on the class of sets A. The limit processes are selfsimilar set-indexed Gaussian processes with continuous sample paths. Using Chentsov's type representations to choose appropriate measures µ and particular sets A, we show that these limits can be Lévy (fractional) Brownian fields or (fractional) Brownian sheets.

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“…Hannan (1960) (page 39, under the i.i.d. assumption for t ), and E. Hannan et al (1972), where the constant C * = 2 ∞ r=−∞ γ 2 (r) + κ 4 ( j ψ 2 j ) 2 , where κ 4 is the 4-th cumulant of t (C < ∞ is implied by the assumptions); an analogous result is implied by Lemma 1 in Wu and Pourahmadi (2009) (taking α = 4), where the constant C * depends on [E(Y 4 t )] 1/4 and the physical dependence measure of the process; see Wu (2011) for details. The same result is obtained by applying the Marcinkiewicz-Zygmund inequality, theorem 4.1 in Dedecker et al (2007); see also Bickel and Gel (2011).…”

Section: A2 Consistency Of the Sample Autocovariancesmentioning

confidence: 97%

A Durbin-Levinson Regularized Estimator of High Dimensional Autocovariance Matrices

Proietti

Giovannelli

2017

SSRN Journal

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We consider the problem of estimating the high-dimensional autocovariance matrix of a stationary random process, with the purpose of out of sample prediction and feature extraction. This problem has received several solutions. In the nonparametric framework, the literature has concentrated on banding and tapering the sample autocovariance matrix. This paper proposes and evaluates an alternative approach, based on regularizing the sample partial autocorrelation function, via a modified Durbin-Levinson algorithm that receives as input the banded and tapered partial autocorrelations and returns a sample autocovariance sequence which is positive definite. We show that the regularized estimator of the autocovariance matrix is consistent and its convergence rates is established. We then focus on constructing the optimal linear predictor and we assess its properties. The computational complexity of the estimator is of the order of the square of the banding parameter, which renders our method scalable for high-dimensional time series. The performance of the autocovariance estimator and the corresponding linear predictor is evaluated by simulation and empirical applications.

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Asymptotic theory for stationary processes

Cited by 105 publications

References 180 publications

Covariance Matrix Estimation in Time Series

Covariance Matrix Estimation in Time Series

Invariance principles for self-similar set-indexed random fields

A Durbin-Levinson Regularized Estimator of High Dimensional Autocovariance Matrices

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