Asymptotic spectral theory for nonlinear time series

By Xiaofeng Shao, Wei Biao Wu. 2006

184 citations recorded by the source corpus.

We consider asymptotic problems in spectral analysis of stationary causal processes. Limiting distributions of periodograms and smoothed periodogram spectral density estimates are obtained and applications to the spectral domain bootstrap are given. Instead of the commonly used strong mixing conditions, in our asymptotic spectral theory we impose conditions only involving (conditional) moments, which are easily verifiable for a variety of nonlinear time series.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ddf79d52bdc8480ae712b001d2f3930272e78091

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Citations in the corpus

HIGHER-ORDER ACCURATE, POSITIVE SEMIDEFINITE ESTIMATION OF LARGE-SAMPLE COVARIANCE AND SPECTRAL DENSITY MATRICES (2005, background): “Remark 2.2 The asymptotic normality of √ T/Sjj(F̂jj − EF̂jj) has been shown under a variety of weak dependence and moment conditions; see e.g. Hannan (1970), Brillinger (1981), Brockwell and Davis (1991), Rosenblatt (1984), Politis and Romano (1995), or Shao and Wu (2007).”
Dynamic Factor Models with Infinite-Dimensional Factor Space: Asymptotic Analysis (2017, result): “Our consistency results for the estimators described in (ii) and (iii) above are based on recent results on lag-window spectral estimators in Shao and Wu (2007) and Liu and Wu (2010), as extended to the multivariate case by Wu and Zaffaroni (2014).”
Komlós–Major–Tusnády approximation under dependence (2014, methodology): “For further examples of processes allowing the representation (1.4), we refer to Wiener (1958), Tong (1990), Priestley (1988), Shao and Wu (2007), Wu (2011) and the examples in Berkes, Hörmann and Schauer (2011).”
Fourier analysis of stationary time series in function space (2013, background): “copy of ε0, Liu and Wu (2010) establish that the spectral density estimator at a fixed frequency is asymptotically Gaussian. The idea behind these coupling conditions is to approximate the series ξt by m-dependent series, for which derivation of asymptotic results is easier. We also mention that, un”
On the range of validity of the autoregressive sieve bootstrap (2012, background): “Recall that under Assumption (A3) it has been shown under different regularity conditions [see Shao and Wu (2007)] that √ nh(fn,X(λ) − fX(λ)) converges to a Gaussian distribution with mean and variance given by the expression on the righthand side of (3.21) and (3.22), respectively.”
Quantile Spectral Processes: Asymptotic Analysis and Inference (2014, unknown): “Proofs and additional examples can be found in Shao and Wu [50] and Shao [49].”
Towards a general theory for nonlinear locally stationary processes (2017, background): “A general theory, however, has been introduced in Wu, W.B. (2005) for Bernoulli shift processes in combination with the functional dependence measure - an important example being Markov processes (and also linear processes). By using this calculus it is possible to transfer a large number of results”
Uniform change point tests in high dimension (2015, background): “ = X |k|≤bn (2.10) ω(k/bn)φbk,h, bn →∞, with weight function ω(x), where φb h,j =(n−j)−1 Xn k=j+1 (Xk,h −Xh)(Xk−j,h −Xh), and Xh = n−1 Pn k=1 Xk,h. Setting ω(x) = 1, we obtain the plain estimate (cf. [45]). For the sake of simplicity, we just consider the plain estimate for our theoretical analysis,”
Inference of trends in time series (2007, background): “By theorem 1 in Wu (2005), under Σ∞i=1 ‖ei − eÅi ‖4 ∞, we have ‖Sn‖4 =O. √ n/.”
Robust Spectral Analysis (2011, background): “The GMC property then follows since ut is GMC by Theorem 5.2 of Shao and Wu (2007).”
BANDING SAMPLE AUTOCOVARIANCE MATRICES OF STATIONARY PROCESSES (2009, background): “1 in Shao and Wu (2007). The latter paper also gives more examples.”
Asymptotic theory for stationary processes (2011, background): “Suitable conditions on R implies GMC. Theorem 2 (Shao and Wu, 2007).”
LOCAL WHITTLE ESTIMATION OF FRACTIONAL INTEGRATION FOR NONLINEAR PROCESSES (2007, background): “…of un+ Here $«t '%t Z is an i+i+d+ copy of $«t %t Z + The property ~9! indicates that the process $un% forgets its past exponentially fast, and it can be verified for many nonlinear time series models ~Wu and Min, 2005; Shao and Wu, 2007!+ Define the fourth cumulant spectral density f4~w1,w2 ,w3 !”
Testing stationarity of functional time series (2014, background): “…similar to those stated above have been used extensively in recent theoretical work, as all stationary time series models in practical use can be represented as Bernoulli shifts, see Wu (2005), Shao and Wu (2007), Aue et al. (2009), Hörmann and Kokoszka (2010), among many other contributions.”