ℓ1-symmetric vector random fields

This paper studies the properties of ${?}_1$-symmetric vector random fields in ${\mathbb{R}}^d$, whose direct/cross covariances are functions of ${?}_1$-norm. The spectral representation and a turning bands expression of the covariance matrix function are derived for an ${?}_1$-symmetric vector random field that is mean square continuous. We also establish an integral relationship between an ${?}_1$-symmetric covariance matrix function and an isotropic one. In addition, a simple but efficient approach is proposed to construct the ${?}_1$-symmetric random field in ${\mathbb{R}}^d$, whose univariate marginal distributions may be taken as arbitrary infinitely divisible distribution with finite variance.     

Student's t Vector Random Fields with Power-Law and Log-Law Decaying Direct and Cross Covariances

This article deals with the Student's t vector random field, which is formulated as a scale mixture of Gaussian vector random fields, and whose finite-dimensional distributions decay in power-law and have heavy tails. There are two classes of Student's t vector random fields, one with second-order moments, and the other without a second-order moment. A Cauchy vector random field is an example of Student's t vector random fields without a first-order moment, and is also an example of Stable vector random fields. A second-order Student's t vector random field allows for any given correlation structure, just as a Gaussian vector random field does. We propose four types of covariance matrix structures for second-order Student's t vector random fields, which decay in power-law or log-law.     

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

We consider anisotropic self-similar random fields, in particular, the fractional Brownian sheet (fBs). This Gaussian field is an extension of fractional Brownian motion. It is well known that the fractional Brownian motion is a unique Gaussian self-similar process with stationary increments. The main result of this article is an example of a Gaussian self-similar field with stationary rectangular increments that is not an fBs. So we proved that the structure of self-similar Gaussian fields can be substantially more involved then the structure of self-similar Gaussian processes. In order to establish the main result, we prove some properties of covariance function for self-similar fields with rectangular increments. Also, using Lamperti transformation, we obtain properties of covariance function for the corresponding stationary fields.