Nonlinear system theory: Another look at dependence

Wu, Wei Biao

doi:10.1073/pnas.0506715102

Cited by 477 publications

(483 citation statements)

References 35 publications

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“…In fact, in this situation, the number of coefficients λ(nA ∩ R j ) which are not 0 or 1 is asymptotically negligible compared to n In dimension d = 1, as stated in Theorem 3 of Wu [33], the result remains true under the weaker condition p = 2. It is a consequence of Corollary 3 in [11] (see also [17]).…”

Section: 4mentioning

confidence: 66%

“…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%

“…Here we adopt the setting of physical dependence measure (p-stability) introduced in Wu [33] in dimension 1 and extended by El Machkouri, Volný and Wu [14] to general dimension d ≥ 1. Several limit theorems are proved under p-stability, see [34] and references therein.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: 4mentioning

confidence: 66%

“…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%

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Invariance principles for self-similar set-indexed random fields

Biermé

Durieu

2014

Trans. Amer. Math. Soc.

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“…However, the currently available exponential inequality associated with NED is not su¢ cient for establishing sharp empirical process results and hence fails to achieve the optimal rates of convergence for general penalized sieve extremum estimators for nonlinear semi-nonparametric models. Andrews (1991b) Another useful dependence measure for strictly stationary nonlinear time series is the so-called physical and predictive dependence measure; see, e.g., Wu (2005Wu ( , 2011. Suppose that fY t g 1 t= 1 is strictly stationary and can be represented as…”

Section: Digression: Nonlinearity and Temporal Dependencementioning

confidence: 99%

Penalized sieve estimation and inference of semi-nonparametric dynamic models: a selective review

Chen

2011

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In this selective review, we …rst provide some empirical examples that motivate the usefulness of semi-nonparametric techniques in modelling economic and …nancial time series. We describe popular classes of semi-nonparametric dynamic models and some temporal dependence properties. We then present penalized sieve extremum (PSE) estimation as a general method for semi-nonparametric models with cross-sectional, panel, time series, or spatial data. The method is especially powerful in estimating di¢ cult ill-posed inverse problems such as semi-nonparametric mixtures or conditional moment restrictions. We review recent advances on inference and large sample properties of the PSE estimators, which include (1) consistency and convergence rates of the PSE estimator of the nonparametric part; (2) limiting distributions of plug-in PSE estimators of functionals that are either smooth (i.e., root-n estimable) or non-smooth (i.e., slower than root-n estimable); (3) simple criterion-based inference for plug-in PSE estimation of smooth or non-smooth functionals; and (4) root-n asymptotic normality of semiparametric two-step estimators and their consistent variance estimators. Examples from dynamic asset pricing, nonlinear spatial VAR, semiparametric GARCH, and copula-based multivariate …nancial models are used to illustrate the general results.

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“…Hannan and Deistler (1988) showed that, for linear ARMA processes and for ≤ (log n) α , α < ∞, the infinity norm ofΓ n, − Γ n is O n −1/2 √ log log n . Wu and Pourahmadi (2009) proved the consistency of (3) for the class of non-linear short-range dependent processes considered by Wu (2005), and obtained an explicit upper bound for the operator norm ofΓ n, −Γ n See Appendix A for a review of the different matrix norms. Bickel and Gel (2011) obtained the consistency of (3) under the Frobenius norm.…”

Section: Introductionmentioning

confidence: 99%

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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Nonlinear system theory: Another look at dependence

Cited by 477 publications

References 35 publications

Invariance principles for self-similar set-indexed random fields

Invariance principles for self-similar set-indexed random fields

Penalized sieve estimation and inference of semi-nonparametric dynamic models: a selective review

A Durbin-Levinson Regularized Estimator of High Dimensional Autocovariance Matrices

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