Limiting Distributions of Non-Linear Vector Functions of Stationary Gaussian Processes.

Ho, Hwai‐Chung; Sun, Tze-Chien

doi:10.21236/ada194569

Cited by 3 publications

(3 citation statements)

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“…), is long range dependent if there exist a slowly varying function at infinity L such that the covariance function γ (p,q) (j) satisfies the relation Example 1. As in the Theorem 3A of Major (1981) and Ho and Sun (1990)…”

Section: Multivariate Gaussian Random Fieldmentioning

confidence: 97%

Parametric Estimation of Long Memory Multivariate Gaussian random fields

N'dri¹,

Kamagaté²,

Hili³

2021

jas

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Section: Multivariate Gaussian Random Fieldmentioning

confidence: 97%

Parametric Estimation of Long Memory Multivariate Gaussian random fields

N'dri¹,

Kamagaté²,

Hili³

2021

jas

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“…Theorem 4 in [1], [3] or the discussion of this problem in a more general setting in the book [14]. It may be worth mentioning also the paper [9], where Ho, H. C. and Sun, T. C. proved an interesting result about the limit distribution of a linear functional of a stationary Gaussian process into R 2 , where the first coordinate of the limit was Gaussian and the second coordinate was non-Gaussian. It may be interesting to find the natural multivariate generalization of this result.…”

Section: Let the Coordinates Gmentioning

confidence: 99%

Wiener--Ito integral representation in vector valued Gaussian stationary random fields

Major

2019

Preprint

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The subject of this work is the multivariate generalization of the theory of multiple Wiener-Itô integrals. In the scalar valued case this theory was described in paper [11]. Our proofs apply the technique of this work, but in the proof of some results new ideas were needed. The motivation for this study was a result in paper [1] of Arcones where he formulated the multivariate version of a non-central limit theorem for non-linear functionals of Gaussian stationary random fields presented in paper [6]. We found the proof in paper [1] incomplete and wanted to give a full proof. We did it in paper [13], but in that proof we needed a detailed description of the properties of non-linear functionals of vector valued stationary Gaussian fields. Here we provide the foundation needed to carry out that proof.We start this study with the investigation of the linear functionals of vector valued Gaussian stationary fields. After an introduction in Section 1 we give the spectral representation of their correlation function in Section 2. Then we construct their random spectral measure and represent the elements of a vector valued Gaussian stationary random field as its Fourier transform in Section 3. We introduce vector valued Gaussian stationary generalized random fields, and prove the analogues of the above results for them in Section 4. We need these results in the study of non-linear functionals of vector valued Gaussian stationary random fields.We start the study of the non-linear functionals with the definition of multiple Wiener-Itô integrals with respect to vector valued random spectral measures in Section 5. Here we also prove some of their important properties. In Section 6 we prove the so-called diagram formula about the presentation of the product of two multiple Wiener-Itô integrals in the form of a sum of multiple Wiener-Itô integrals. This result is applied in Section 7 in the proof of the multidimensional version of Itô's formula which creates an important relation between multiple Wiener-Itô integrals and Wick polynomials. Wick polynomials are the multivariate versions of Hermite polynomials. Another important result of Section 7 is a formula about the calculation of the shift transforms of a random variable given in the form of a sum of multiple Wiener-Itô integrals. In Section 8 we work

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“…Example 1. As in the Theorem 3A of Major (1981) and Ho and Sun (1990) there are measures G p,q , for 1 ≤ p, q ≤ d such that…”

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confidence: 99%