The analysis of two-dimensional stationary processes with discontinuous spectrat

Abstract

It is well-known that the spectral theory of one-dimensional stationary processes may be generalized to the case of multidimensional stationary processes. In particular, Bartlett (1955, p. 191) considered two-dimensional processes and derived a spectral representation in terms of a doubly orthogonal process. Multidimensional processes arise naturally in practice whenever we consider 'fields' whose spatial properties we wish to study, in addition to their variations with time. For example, in the statistical theory of turbulence, each component of the velocity vector may be regarded as a four-dimensional process, depending on the three-space co-ordinates and the time co-ordinates (Bartlett, 1955, p. 193), while Longuet-Higgins (1957) used a two-dimensional process to describe the behaviour of sea waves. Pierson & Tick (1957) considered two-dimensional processes in meteorology and oceanography, and a similar model has been suggested for the analysis of waves in a paper mill. Let Xt, T(t = O, ? 1, + 2, ... .; = 0, ? 1, ? 2, ...) be a discrete two-dimensional stationary process. The autocovariance function, R8, u, is defined by u = ~~~~~~~~~~~~~(1.1) RS, ,,=EfXt, T XtJrs, +?l (1 assuming E{X 0 j = 0, all t, r and letting * denote the complex conjugate. There exists a spectral representation of RB u in the form