Weakly Stationary Point Processes and Random Measures

Abstract

Summary A weakly stationary point process on the real line is a point process with counting measure N(A) defined on Borel subsets A of the real line such that the first two moments of N(A) are finite for bounded Borel sets and invariant under translation. The expected number of points falling in an interval of length u equals μu and serves to define the intensity parameter μ of the process. The variance of the number of points falling in an interval of length u defines the variance function V(u) which, with μ, characterizes weakly stationary point processes. V(.) is absolutely continuous with its left- and right-hand derivatives differing on at most a countable dense set, with V′(0 +)= α=lim⁡θ→∞πG([O,θ])/θ and G(.) a uniquely determined σ-finite spectral measure on [0, ∞) in terms of which V(u)=∫[0,∞)(sin 12θu/12θ)2 G(dθ). These general results are also true of any weakly stationary signed random measure. To any operation on a point process there corresponds an operation on the spectral measure, and the analogues of superposition, independent random motion of the points, deletion and multiplication of points and of forming cluster processes are exhibited. The paper attempts to expound a rigorous basis for the general theory preceding Bartlett's (1963) development of spectral analysis for point processes.