'The spectral analysis of two-dimensional point processes.'

Abstract

1. BASIC FORMULAE In a previous paper (Bartlett, 1963 b) I have developed in some detail statistical techniques for estimating and analysing the spectrum of a stationary point stochastic process in one dimension such as time. In a constructive discussion on this paper by various contributors, an interesting extension was suggested by Dr G. M. Jenkins to the bivariate case (either a bivariate point process, or a mixed bivariate, with one point process component and one continuous component). As with continuous processes, point processes can be specified in more than one dimension; and the extension I myself had noted with a view to further development was to the multi-dimensional case in this latter sense. The present paper considers the spectral analysis of two-dimensional stationary* point processes (e.g. two Cartesian space co-ordinates x and y, denoted by the vector r). The two-dimensional stationary point process N(r), where N(r) denotes the cumulative number of points at r, is assumed to have the properties: