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16
Estimators of fractal dimension: Assessing the roughness of time series and spatial data
, 2010
"... Lies, damn lies, and dimension estimates ..."
Bipower variation for Gaussian processes with stationary increments
"... Convergence in probability and central limit laws of bipower variation for Gaussian processes with stationary increments and for integrals with respect to such processes are derived. The main tools of the proofs are some recent powerful techniques of Wiener/Itô/Malliavin calculus for establishing li ..."
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Cited by 12 (1 self)
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Convergence in probability and central limit laws of bipower variation for Gaussian processes with stationary increments and for integrals with respect to such processes are derived. The main tools of the proofs are some recent powerful techniques of Wiener/Itô/Malliavin calculus for establishing limit laws, due to Nualart, Peccati and others.
Multipower variation for Brownian semistationary processes (full version). CREATES research paper 200921, Aarhus Univ. Available at http://www.econ.au.dk/research/researchcentres/creates/research/researchpapers/ researchpapers2009/. Asymptotic result
, 2009
"... In this paper we study the asymptotic behaviour of power and multipower variations of processes Y : where g : (0, ∞) → R is deterministic, σ > 0 is a random process, W is the stochastic Wiener measure and Z is a stochastic process in the nature of a drift term. Processes of this type serve, in p ..."
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Cited by 11 (9 self)
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In this paper we study the asymptotic behaviour of power and multipower variations of processes Y : where g : (0, ∞) → R is deterministic, σ > 0 is a random process, W is the stochastic Wiener measure and Z is a stochastic process in the nature of a drift term. Processes of this type serve, in particular, to model data of velocity increments of a fluid in a turbulence regime with spot intermittency σ . The purpose of this paper is to determine the probabilistic limit behaviour of the (multi)power variations of Y as a basis for studying properties of the intermittency process σ . Notably the processes Y are in general not of the semimartingale kind and the established theory of multipower variation for semimartingales does not suffice for deriving the limit properties. As a key tool for the results, a general central limit theorem for triangular Gaussian schemes is formulated and proved. Examples and an application to the realised variance ratio are given.
Quantitative BreuerMajor Theorems
 Stoch. Proc. App
, 2011
"... We consider sequences of random variables of the type Sn = n−1/2 ∑n k=1 {f(Xk)− E[f(Xk)]}, n ≥ 1, where X = (Xk)k∈Z is a ddimensional Gaussian process and f: Rd → R is a measurable function. It is known that, under certain conditions on f and the covariance function r of X, Sn converges in distribu ..."
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Cited by 8 (3 self)
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We consider sequences of random variables of the type Sn = n−1/2 ∑n k=1 {f(Xk)− E[f(Xk)]}, n ≥ 1, where X = (Xk)k∈Z is a ddimensional Gaussian process and f: Rd → R is a measurable function. It is known that, under certain conditions on f and the covariance function r of X, Sn converges in distribution to a normal variable S. In the present paper we derive several explicit upper bounds for quantities of the type E[h(Sn)] − E[h(S)], where h is a su ciently smooth test function. Our methods are based on Malliavin calculus, on interpolation techniques and on the Stein's method for normal approximation. The bounds deduced in our paper depend only on Var[f(X1)] and on simple in nite series involving the components of r. In particular, our results generalize and re ne some classic CLTs by BreuerMajor, GiraitisSurgailis and Arcones, concerning the normal approximation of partial sums associated with
Asymptotic theory for Brownian semistationary processes with application to turbulence
, 2014
"... This paper presents some asymptotic results for statistics of Brownian semistationary (BSS) processes. More precisely, we consider power variations of BSS processes, which are based on high frequency (possibly higher order) differences of the BSS model. We review the limit theory discussed in [4, ..."
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Cited by 4 (4 self)
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This paper presents some asymptotic results for statistics of Brownian semistationary (BSS) processes. More precisely, we consider power variations of BSS processes, which are based on high frequency (possibly higher order) differences of the BSS model. We review the limit theory discussed in [4, 5] and present some new connections to fractional diffusion models. We apply our probabilistic results to construct a family of estimators for the smoothness parameter of the BSS process. In this context we develop estimates with gaps, which allow to obtain a valid central limit theorem for the critical region. Finally, we apply our statistical theory to turbulence data.
Limit theorems for power variations of ambit fields driven by white noise
, 2013
"... We study the asymptotic behavior of lattice power variations of twoparameter ambit fields that are driven by white noise. Our first result is a law of large numbers for such power variations. Under a constraint on the memory of the ambit field, normalized power variations are shown to converge to c ..."
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Cited by 3 (1 self)
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We study the asymptotic behavior of lattice power variations of twoparameter ambit fields that are driven by white noise. Our first result is a law of large numbers for such power variations. Under a constraint on the memory of the ambit field, normalized power variations are shown to converge to certain integral functionals of the volatility field associated to the ambit field, when the lattice spacing tends to zero. This law of large numbers holds also for thinned power variations that are computed by only including increments that are separated by gaps with a particular asymptotic behavior. Our second result is a related stable central limit theorem for thinned power variations. Additionally, we provide concrete examples of ambit fields that satisfy the assumptions of our limit theorems.
Ambit fields: survey and new challenges
, 2014
"... In this paper we present a survey on recent developments in the study of ambit fields and point out some open problems. Ambit fields is a class of spatiotemporal stochastic processes, which by its general structure constitutes a flexible model for dynamical structures in time and/or in space. We wi ..."
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In this paper we present a survey on recent developments in the study of ambit fields and point out some open problems. Ambit fields is a class of spatiotemporal stochastic processes, which by its general structure constitutes a flexible model for dynamical structures in time and/or in space. We will review their basic probabilistic properties, main stochastic integration concepts and recent limit theory for high frequency statistics of ambit fields.