Example of a Gaussian Self-Similar Field With Stationary Rectangular Increments That Is Not a Fractional Brownian Sheet

Makogin, Vitalii; Mishura, Yuliya

doi:10.1080/07362994.2014.1002042

Cited by 5 publications

(6 citation statements)

References 9 publications

(14 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A fractional Brownian sheet has strictly stationary rectangular increments. It is proved in [2] but this fact can be also derived from representation (16…”

Section: Gaussian Self-similar Random Fields With Strictly Stationarymentioning

confidence: 94%

“…which is the version of the right hand side of (59) in the case H = (0.5, 0.5). From [16] follows that Y H/2 is self-similar and has mild stationary rectangular increments, i.e., Y 1/2 ∈ C H,2 M . To show that rectangular increments of Y 1/2 are not wide-sense stationary we write down their covariance function.…”

Section: Spectral Representation Of the Fractional Brownian Motionmentioning

confidence: 98%

“…We look for the density g(x), x ∈ R in the form of the inverse Fourier transform of C f Bs . In the paper [16] it is showed that C f Bs is integrable. Indeed,…”

Section: Spectral Representation Of the Fractional Brownian Motionmentioning

confidence: 99%

“…With the help of these representations we construct various examples of fields from C H,N S , N ≥ 2 which are not fractional Brownian sheets. In our previous paper [16], we have provided such example for the class C H,N M , N ≥ 2, namely, we have proved that C H,N M , N ≥ 2 contains not only the fractional Brownian sheets. In the present paper, we describe the whole class C H,N M using a Lamperti transformation and a spectral representation of the stationary Gaussian random fields.…”

mentioning

confidence: 95%

See 3 more Smart Citations

Gaussian multi-self-similar random fields with distinct stationary properties of their rectangular increments

Makogin

Mishura

2019

Stochastic Models

Self Cite

View full text Add to dashboard Cite

We describe two classes of Gaussian self-similar random fields: with strictly stationary rectangular increments and with mild stationary rectangular increments. We find explicit spectral and moving average representations for the fields with strictly stationary rectangular increments and characterize fields with mild stationary rectangular increments by the properties of covariance functions of their Lamperti transformations as well as in terms of their spectral densities. We establish that both classes contain not only fractional Brownian sheets and we provide corresponding examples. As a by-product, we obtain a new spectral representation for the fractional Brownian motion.

show abstract

“…A fractional Brownian sheet has strictly stationary rectangular increments. It is proved in [2] but this fact can be also derived from representation (16…”

Section: Gaussian Self-similar Random Fields With Strictly Stationarymentioning

confidence: 94%

Section: Spectral Representation Of the Fractional Brownian Motionmentioning

confidence: 98%

“…We look for the density g(x), x ∈ R in the form of the inverse Fourier transform of C f Bs . In the paper [16] it is showed that C f Bs is integrable. Indeed,…”

Section: Spectral Representation Of the Fractional Brownian Motionmentioning

confidence: 99%

mentioning

confidence: 95%

See 2 more Smart Citations

Gaussian multi-self-similar random fields with distinct stationary properties of their rectangular increments

Makogin

Mishura

2019

Stochastic Models

Self Cite

View full text Add to dashboard Cite

show abstract

“…So several different classes of anisotropic Gaussian random fields such as fractional Brownian sheets have been introduced for theoretical and application purposes and some sample-function behavior of them studied [29], [30]. In the following, we present the definitions of centered Gaussian random field as the fractional Brownian sheet and the stationary rectangular increments property [16]. Definition 6.…”

Section: Theoretical Structurementioning

confidence: 99%

Multi-scale Invariant Fields: Estimation and Prediction

Ghasemi¹,

Rezakhah²,

Modarresi³

2017

Preprint

View full text Add to dashboard Cite

Extending the concept of multi-selfsimilar random field we study multi-scale invariant (MSI) fields which have component-wise discrete scale invariant property. Assuming scale parameters as λ i > 1, i = 1, . . . , d and the parameter space as (1, ∞) d , the first scale rectangle is referred to the rectangle (1, λ 1 ) × . . . × (1, λ d ). Applying certain component-wise geometric sampling of MSI field, the harmonic-like representation and spectral density of the sampled MSI field are characterized. Furthermore, the covariance function and spectral density of the sampled Markov MSI field are presented by the variances and covariances of samples inside first scale rectangle. As an example of MSI field, a two-dimensional simple fractional Brownian sheet (sfBs) is demonstrated. Also real data of the precipitation in some area of Brisbane in Australia for two days (25 and 26 January 2013) are examined. We show that precipitation on this area has MSI property and estimate it as a simple MSI field with stationary increments inside scale intervals. This structure enables us to predict the precipitation in surface and time. We apply the mean absolute percentage error as a measure for the accuracy of the predictions.

show abstract

On stationarity properties of generalized Hermite-type processes

Donhauzer

Olenko

2020

Stochastics

View full text Add to dashboard Cite

Example of a Gaussian Self-Similar Field With Stationary Rectangular Increments That Is Not a Fractional Brownian Sheet

Cited by 5 publications

References 9 publications

Gaussian multi-self-similar random fields with distinct stationary properties of their rectangular increments

Gaussian multi-self-similar random fields with distinct stationary properties of their rectangular increments

Multi-scale Invariant Fields: Estimation and Prediction

On stationarity properties of generalized Hermite-type processes

Contact Info

Product

Resources

About