Martingale-coboundary representation for a class of random fields

It is known that under some conditions, a stationary random sequence admits a representation as a sum of two sequences: one of them is a martingale difference sequence, and another one is a so-called coboundary. Such a representation can be used for proving some limit theorems by means of the martingale approximation. A multivariate version of such a decomposition is presented in the paper for a class of random fields generated by several commuting, noninvertible, probability preserving transformations. In this representation, summands of mixed type appear, which behave with respect to some group of directions of the parameter space as reversed multiparameter martingale differences (in the sense of one of several known definitions), while they look as coboundaries relative to other directions. Applications to limit theorems will be published elsewhere. Bibliography: 14 titles. © 2009 Springer Science+Business Media, Inc.