scholarly journals Representations of isotropic Gaussian random fields with homogeneous increments

2006 ◽  
Vol 2006 ◽  
pp. 1-25 ◽  
Author(s):  
Kacha Dzhaparidze ◽  
Harry van Zanten ◽  
Pawel Zareba
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Random Fields ◽  
Moving Average ◽  
Gaussian Random Fields ◽  
Present Series ◽  
Series Expansions

We present series expansions and moving average representations of isotropic Gaussian random fields with homogeneous increments, making use of concepts of the theory of vibrating strings. We illustrate our results using the example of Lévy's fractional Brownian motion on ℝN.

scholarly journals Limit laws on extremes of nonhomogeneous Gaussian random fields

2017 ◽  
Vol 54 (3) ◽  
pp. 811-832 ◽  
Author(s):  
Zhongquan Tan
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Random Fields ◽  
Gaussian Random Fields ◽  
Tail Asymptotics ◽  
Limit Laws ◽  
Integrated Processes

Abstract In this paper, by using the tail asymptotics derived by Dębicki et al. (2016), we prove the Gumbel limit laws for the maximum of a class of nonhomogeneous Gaussian random fields. As an application of the main results, we derive the Gumbel limit law for Shepp statistics of fractional Brownian motion and Gaussian integrated processes.

A central limit theorem for super-Brownian motion with super-Brownian immigration

1999 ◽  
Vol 36 (4) ◽  
pp. 1218-1224 ◽  
Author(s):  
Wen-Ming Hong ◽  
Zeng-Hu Li
Keyword(s):  
Brownian Motion ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Fields ◽  
Gaussian Random Fields ◽  
Potential Operator ◽  
Super Brownian Motion

We prove a central limit theorem for the super-Brownian motion with immigration governed by another super-Brownian. The limit theorem leads to Gaussian random fields in dimensions d ≥ 3. For d = 3 the field is spatially uniform; for d ≥ 5 its covariance is given by the potential operator of the underlying Brownian motion; and for d = 4 it involves a mixture of the two kinds of fluctuations.

A note on two-level superprocesses

2004 ◽  
Vol 41 (1) ◽  
pp. 202-210
Author(s):  
Wen-Ming Hong
Keyword(s):  
Brownian Motion ◽  
Central Limit ◽  
Random Fields ◽  
Limit Theorems ◽  
Gaussian Random Fields ◽  
Central Limit Theorems ◽  
Gaussian Fields ◽  
Super Brownian Motion ◽  
Random Immigration

We prove some central limit theorems for a two-level super-Brownian motion with random immigration, which lead to limiting Gaussian random fields. The covariances of those Gaussian fields are explicitly characterized.

A central limit theorem for super-Brownian motion with super-Brownian immigration

1999 ◽  
Vol 36 (04) ◽  
pp. 1218-1224 ◽  
Author(s):  
Wen-Ming Hong ◽  
Zeng-Hu Li
Keyword(s):  
Brownian Motion ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Fields ◽  
Gaussian Random Fields ◽  
Potential Operator ◽  
Super Brownian Motion

We prove a central limit theorem for the super-Brownian motion with immigration governed by another super-Brownian. The limit theorem leads to Gaussian random fields in dimensions d ≥ 3. For d = 3 the field is spatially uniform; for d ≥ 5 its covariance is given by the potential operator of the underlying Brownian motion; and for d = 4 it involves a mixture of the two kinds of fluctuations.

scholarly journals A note on Lévy’s Brownian motion II

1989 ◽  
Vol 114 ◽  
pp. 165-172 ◽  
Author(s):  
Si Si
Keyword(s):  
Brownian Motion ◽  
Random Fields ◽  
Probability Distributions ◽  
Joint Probability ◽  
Variational Calculus ◽  
Normal Derivative ◽  
Gaussian Random Fields ◽  
Normal Direction ◽  
Joint Probability Distributions ◽  
Derivatives Of

The purpose of this paper is to discuss some particular random fields derived from Lévy’s Brownian motion to find its characteristic properties of the joint probability distributions. In [9], special attention was paid to the behaviour of the Brownian motion when the parameter runs along a curve in the parameter space, and with this property the conditional expectation has been obtained when the values are known on the curve.The present paper deals with the variation of the Brownian motion in the normal direction to a given curve, in contrast to the case in [9], where we discussed the properties along the curve. Actually we shall find, in this paper, formulae of the variation with the help of the normal derivative of Brownian motion and observe its singularity. We then discuss partial derivatives of Rd-parameter Lévy’s Brownian motion and make attempt to restrict the parameter to a hypersurface so that we obtain new random fields on that hypersurface. By comparing such derivatives with those of other Gaussian random fields, we can see that the singularity of the new random fields seems to be an interesting characteristic of Lévy’s Brownian motion. Further, we hope that our approach may be thought of as a first step to the variational calculus for Gaussian random fields.

scholarly journals A note on moving average models for Gaussian random fields

2013 ◽  
Vol 83 (3) ◽  
pp. 850-855 ◽  
Author(s):  
Linda V. Hansen ◽  
Thordis L. Thorarinsdottir
Keyword(s):  
Random Fields ◽  
Moving Average ◽  
Gaussian Random Fields ◽  
Moving Average Models
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