Bipower Variation for Gaussian Processes with Stationary Increments

Barndorff–Nielsen, Ole E.; Corcuera, José Manuel; Podolskij, Mark; Woerner, Jeannette H. C.

doi:10.1017/s0021900200005271

Cited by 19 publications

(30 citation statements)

References 31 publications

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“…In this article we have employed techniques that were successfully used in the univariate case for the power, multipower, and bipower variation of the BSS process and of Gaussian processes, (as appearing in Barndorff-Nielsen et al (2011), Barndorff-Nielsen, , Barndorff-Nielsen, Corcuera, Podolskij and Woerner (2009), Corcuera et al (2013)) to show a central limit theorem for the realised covariation of the bivariate Gaussian core and the BSS process.…”

Section: Resultsmentioning

confidence: 99%

A central limit theorem for the realised covariation of a bivariate Brownian semistationary process

Granelli¹,

Veraart²

2019

Bernoulli

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This article presents a weak law of large numbers and a central limit theorem for the scaled realised covariation of a bivariate Brownian semistationary process. The novelty of our results lies in the fact that we derive the suitable asymptotic theory both in a multivariate setting and outside the classical semimartingale framework. The proofs rely heavily on recent developments in Malliavin calculus.

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Section: Resultsmentioning

confidence: 99%

A central limit theorem for the realised covariation of a bivariate Brownian semistationary process

Granelli¹,

Veraart²

2019

Bernoulli

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“…In the last decade, asymptotic theory for power variations of various classes of stochastic processes has been intensively investigated in the literature. We refer, for example, to [5,24,25,31] for limit theory for power variations of Itô semimartingales, to [3,4,17,21,30] for the asymptotic results in the framework of fractional Brownian motion and related processes, and to [15,16,38] for investigations of power variation of the Rosenblatt process.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Power variation for a class of stationary increments Lévy driven moving averages

Basse-O'Connor¹,

Podolskij²

2017

Ann. Probab.

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In this paper, we present some new limit theorems for power variation of kth order increments of stationary increments Lévy driven moving averages. In the infill asymptotic setting, where the sampling frequency converges to zero while the time span remains fixed, the asymptotic theory gives novel results, which (partially) have no counterpart in the theory of discrete moving averages. More specifically, we show that the first-order limit theory and the mode of convergence strongly depend on the interplay between the given order of the increments k ≥ 1, the considered power p > 0, the BlumenthalGetoor index β ∈ [0, 2) of the driving pure jump Lévy process L and the behaviour of the kernel function g at 0 determined by the power α. First-order asymptotic theory essentially comprises three cases: stable convergence towards a certain infinitely divisible distribution, an ergodic type limit theorem and convergence in probability towards an integrated random process. We also prove a second-order limit theorem connected to the ergodic type result. When the driving Lévy process L is a symmetric β-stable process, we obtain two different limits: a central limit theorem and convergence in distribution towards a (k − α)β-stable totally right skewed random variable.

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“…We refer e.g. to [4,12,13,15] for limit theory for power variations of Itô semimartingales, to [2,3,9,11,14] for the asymptotic results in the framework In a recent paper [5] the power variation of stationary increments Lévy driven moving averages has been studied. Let us recall the definitions, notations and main results of this paper.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On critical cases in limit theory for stationary increments Lévy driven moving averages

Basse-O'Connor

Podolskij²

2016

Stochastics

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In this paper we present some limit theorems for power variation of stationary increments Lévy driven moving averages in the setting of critical regimes. In [5] the authors derived first and second order asymptotic results for k-th order increments of stationary increments Lévy driven moving averages. The limit theory heavily depends on the interplay between the given order of the increments, the considered power, the Blumenthal-Getoor index of the driving pure jump Lévy process L and the behaviour of the kernel function g at 0. In this work we will study the critical cases, which were not covered in the original work [5].

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Bipower Variation for Gaussian Processes with Stationary Increments

Cited by 19 publications

References 31 publications

A central limit theorem for the realised covariation of a bivariate Brownian semistationary process

A central limit theorem for the realised covariation of a bivariate Brownian semistationary process

Power variation for a class of stationary increments Lévy driven moving averages

On critical cases in limit theory for stationary increments Lévy driven moving averages

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