Formula omitted]-symmetric vector random fields

Abstract

This paper studies the properties of ℓ1-symmetric vector random fields in Rd, whose direct/cross covariances are functions of ℓ1-norm. The spectral representation and a turning bands expression of the covariance matrix function are derived for an ℓ1-symmetric vector random field that is mean square continuous. We also establish an integral relationship between an ℓ1-symmetric covariance matrix function and an isotropic one. In addition, a simple but efficient approach is proposed to construct the ℓ1-symmetric random field in Rd, whose univariate marginal distributions may be taken as arbitrary infinitely divisible distribution with finite variance.