A central limit theorem for non-instantaneous filters of a stationary Gaussian process

Ho, Chung-Hwai; Sun, Tze-Chien

doi:10.1016/0047-259x(87)90082-0

Cited by 31 publications

(36 citation statements)

References 5 publications

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“…After expressing the functional Itx o > o] as a multiple Wiener-It6 integral expansion and expressing products of expansions through the Diagram Theorem, we obtain the first Proposition and Corollary. C=+l,(mZj~Jdp ~(l_p2)m_j+l/2. In the following theorem, asymptotic normality has been proved by Ho and Sun (1987) and also follows easily from Theorems 1 and 2 of Chambers and Slud (1989a) using the representation of Proposition 1. The positive lower bound for the asymptotic variance is apparently new.…”

Section: Representation Results and Limit Theoremsmentioning

confidence: 93%

Multiple Wiener-itô integral expansions for level-crossing-count functionals

Slud

1991

Probab. Th. Rel. Fields

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Summary. This paper applies the stochastic calculus of multiple Wiener-It6 integral expansions to express the number of crossings of the mean level by a stationary (discrete-or continuous-time) Gaussian process within a fixed time interval [0, T]. The resulting expansions involve a class of hypergeometric functions, for which recursion and differential relations and some asymptotic properties are derived. The representation obtained for level-crossing counts is applied to prove a central limit theorem of Cuzick (1976) for level crossings in continuous time, using a general central limit theorem of Chambers and Slud (1989a) for processes expressed via multiple Wiener-It6 integral expansions in terms of a stationary Gaussian process. Analogous results are given also for discrete-time processes. This approach proves that the limiting variance is strictly positive, without additional assumptions needed by Cuzick.

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Section: Representation Results and Limit Theoremsmentioning

confidence: 93%

Multiple Wiener-itô integral expansions for level-crossing-count functionals

Slud

1991

Probab. Th. Rel. Fields

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“…Ho and Sun [23] have shown a non-functional version of Theorem 2 for statistics of the type (3.12) when the correlation function r n does not depend on n. To the best of our knowledge, Theorem 2 is the first central limit theorem for (general) multipower variation of a row-wise stationary Gaussian process.…”

Section: Theoremmentioning

confidence: 97%

Multipower Variation for Brownian Semistationary Processes

2009

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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.

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“…This condition is often used by authors in relation to limit theorems of stochastic processes [3], [18], [20], [23]. Here, the choice ρ = 2H , 0 < H < 1, is suggested by the form of the variances of self-similar processes.…”

Section: S Albeverio and S Kawasakimentioning

confidence: 99%

“…There are many known results about limit theorems for functionals of general stationary Gaussian vectors (see [3], [4], [18], and [20], for example). However, the important point here is how to adapt the wavelet coefficients to those general results, because (s J 0 +1 (k), .…”

Section: Lemma 2 For Each Pairmentioning

confidence: 99%

On a localization property of wavelet coefficients for processes with stationary increments, and applications. I. Localization with respect to shift

Albeverio

Kawasaki

2005

Adv. Appl. Probab.

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We formulate a localization property of wavelet coefficients for processes with stationary increments, in the estimation problem associated with the processes. A general setting for the estimation is adopted and examples that fit this setting are given. An evaluation of wavelet coefficient decay with respect to shift k ∈ N is explicitly derived (only the asymptotic behavior, for large k, was previously known). It is this evaluation that makes it possible to establish the localization property of the wavelet coefficients. In doing so, it turns out that the theory of positive-definite functions plays an important role. As applications, we show that, in the wavelet coefficient domain, estimators that use a simple moment method are nearly as good as maximum likelihood estimators. Moreover, even though the underlying process is long-range dependent and process domain estimates imply the validity of a noncentral limit theorem, for the wavelet coefficient domain estimates we always obtain a central limit theorem with a small prescribed error.

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A central limit theorem for non-instantaneous filters of a stationary Gaussian process

Cited by 31 publications

References 5 publications

Multiple Wiener-itô integral expansions for level-crossing-count functionals

Multiple Wiener-itô integral expansions for level-crossing-count functionals

Multipower Variation for Brownian Semistationary Processes

On a localization property of wavelet coefficients for processes with stationary increments, and applications. I. Localization with respect to shift

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