Characterization theorems for some classes of covariance functions associated to vector valued random fields

We characterize some important classes of cross-covariance functions associated to vector valued random fields based on latent dimensions. We also give some results for mixture based models that allow for the construction of new cross-covariance models. In particular, we give a criterion for the permissibility of quasi-arithmetic operators in order to construct valid cross covariances.

[1]  T. Gneiting,et al.  Matérn Cross-Covariance Functions for Multivariate Random Fields , 2010 .

[2]  Emilio Porcu,et al.  From Schoenberg to Pick–Nevanlinna: Toward a complete picture of the variogram class , 2008, 0812.2936.

[3]  D. Widder,et al.  The Laplace Transform , 1943, The Mathematical Gazette.

[4]  E. Stein,et al.  Introduction to Fourier Analysis on Euclidean Spaces. , 1971 .

[5]  Jing Guo,et al.  Construction and application of covariance functions with variable length‐fields , 2006 .

[6]  Jorge Mateu,et al.  Quasi-arithmetic means of covariance functions with potential applications to space-time data , 2006, J. Multivar. Anal..

[7]  Josip Pečarić,et al.  Some Inequalities for Monotone Functions , 1993 .

[8]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[9]  Christian Berg,et al.  Potential Theory on Locally Compact Abelian Groups , 1975 .

[10]  T. Gneiting.  Nonseparable, Stationary Covariance Functions for Space–Time Data , 2002 .

[11]  Tatiyana V. Apanasovich,et al.  Cross-covariance functions for multivariate random fields based on latent dimensions , 2010 .

[12]  Michael Scheuerer,et al.  Covariance Models for Divergence-Free and Curl-Free Random Vector Fields , 2012 .

[13]  M. Goulard,et al.  Linear coregionalization model: Tools for estimation and choice of cross-variogram matrix , 1992 .

[14]  J. Mateu,et al.  Nonseparable stationary anisotropic space–time covariance functions , 2006 .

[15]  Martin Schlather,et al.  Some covariance models based on normal scale mixtures , 2011 .

[16]  Harald Cramer,et al.  On the Theory of Stationary Random Processes , 1940 .

[17]  I. J. Schoenberg,et al.  Metric spaces and positive definite functions , 1938 .

[18]  Emilio Porcu,et al.  Characterization theorems for the Gneiting class of space-time covariances , 2011 .

[19]  N. Vakhania,et al.  Probability Distributions on Banach Spaces , 1987 .

[20]  Harrison H. Barrett,et al.  3 – Theory of Random Processes , 1981 .