One-to-one correspondences between discrete multivariate stationary, self-similar, and stationary increment fields

Abstract

In this article, we consider three important classes of n-variate fields indexed by the set of N dimensional integers, namely stationary, stationary increment, and self-similar fields. We connect these classes through bijective transformations. The one-to-one correspondence between stationary and self-similar fields, where the index of self-similarity is a tuple of positive definite matrices, is given by a version of the Lamperti transformation. In addition, we introduce generalized AR(1) type equations, whose unique stationary solutions are obtained via these transformations. Last, we apply the transformations in order to construct multivariate stationary fractional Ornstein-Uhlenbeck fields of the first and second kind, including a brief simulation study of bivariate Ornstein-Uhlenbeck sheets.