On weak invariance principles for sums of dependent random functionals

Jirak, Moritz

doi:10.1016/j.spl.2013.06.014

Cited by 18 publications

(17 citation statements)

References 6 publications

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“…In this case assumption (1.2) follows from the joint measurability of the ϵ i (t, ω)'s. Assumption (1.3) is stronger than the requirement  ∞ ℓ=1 (E∥η j − η j,ℓ ∥ 2 ) 1/2 < ∞ used by Hörmann and Kokoszka (2010), Berkes et al (2013) and Jirak (2013) to establish the central limit theorem for sums of Bernoulli shifts. Since we need the central limit theorem for sample correlations, higher moment conditions and a faster rate in the approximability with ℓ-dependent sequences are needed.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Testing for independence between functional time series

Horváth

Rice

2015

Journal of Econometrics

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Article history:Available online xxxx JEL classification: C12 C32Keywords: Functional observations Tests for independence Weak dependence Long run covariance function Central limit theorem a b s t r a c t Frequently econometricians are interested in verifying a relationship between two or more time series. Such analysis is typically carried out by causality and/or independence tests which have been well studied when the data is univariate or multivariate. Modern data though is increasingly of a high dimensional or functional nature for which finite dimensional methods are not suitable. In the present paper we develop methodology to check the assumption that data obtained from two functional time series are independent. Our procedure is based on the norms of empirical cross covariance operators and is asymptotically validated when the underlying populations are assumed to be in a class of weakly dependent random functions which include the functional ARMA, ARCH and GARCH processes.

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Section: Introduction and Resultsmentioning

confidence: 99%

Testing for independence between functional time series

Horváth

Rice

2015

Journal of Econometrics

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“…We note that Assumption 3.4 implies that e i , −∞ < i < ∞, is a stationary sequence and Assumptions 2.3 and 2.4 are also satisfied (cf. Berkes et al (2013) and Jirak (2013)). To get the exact limit of C N (t, s) under H A we need a further regularity condition:…”

Section: 2mentioning

confidence: 98%

Change point detection in heteroscedastic time series

Górecki

Horváth

Kokoszka

2018

Econometrics and Statistics

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Abstract. Many time series exhibit changes both in level and in variability. Generally, it is more important to detect a change in the level, and changing or smoothly evolving variability can confound existing tests. This paper develops a framework for testing for shifts in the level of a series which accommodates the possibility of changing variability. The resulting tests are robust both to heteroskedasticity and serial dependence. They rely on a new functional central limit theorem for dependent random variables whose variance can change or trend in a substantial way. This new result is of independent interest as it can be applied in many inferential contexts applicable to time series. Its application to change point tests relies on a new approach which utilizes Karhunen-Loéve expansions of the limit Gaussian processes. After presenting the theory in the most commonly encountered setting of the detection of a change point in the mean, we show how it can be extended to linear and nonlinear regression. Finite sample performance is examined by means of a simulation study and an application to yields on US treasury bonds.

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“…Lemma 8.1 comes as a byproduct of the results in [44], see also Lemma 10.3 and [63] for the original argument for real-valued sequences, which we also use in the sequel. As a next result, we state a special type of Höffding decomposition.…”

Section: Proofs Of Sectionmentioning

confidence: 99%

Optimal eigen expansions and uniform bounds

Jirak

2015

Probab. Theory Relat. Fields

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Let X k k∈Z ∈ L 2 (T ) be a stationary process with associated lag operators C h . Uniform asymptotic expansions of the corresponding empirical eigenvalues and eigenfunctions are established under almost optimal conditions on the lag operators in terms of the eigenvalues (spectral gap). In addition, the underlying dependence assumptions are optimal in a certain sense, including both short and long memory processes. This allows us to study the relative maximum deviation of the empirical eigenvalues under very general conditions. Among other things, convergence to an extreme value distribution is shown. We also discuss how the asymptotic expansions transfer to the long-run covariance operator G in a general framework.

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On weak invariance principles for sums of dependent random functionals

Cited by 18 publications

References 6 publications

Testing for independence between functional time series

Testing for independence between functional time series

Change point detection in heteroscedastic time series

Optimal eigen expansions and uniform bounds

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