SUMMAKYIn recent years methods of data analysis for point processes have received some attention, for example, by Cox & Lewis (1966) and Lewis (1964). In particular Bartlett (1963a, b) has introduced methods of analysis based on the point spectrum. Theoretical models are relatively sparse. In this paper the theoretical properties of a class of processes with particular reference to the point spectrum or corresponding covariance density functions are discussed. A particular result is a self-exciting process with the same second-order properties as a certain doubly stochastic process. These are not distinguishable by methods of data analysis based on these properties.at University College London on July 1, 2014 http://biomet.oxfordjournals.org/ Downloaded from J -oo Applying (2), we find, for r > 0, li(j) = A<7(T)+ g{T-v)(i{v)dv.

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Summary The point spectral matrix is obtained for a class of mutually exciting point processes. The solution makes use of methods similar to those used in solving Wiener–Hopf integral equations.

It is shown that all stationary self-exciting point processes with finite intensity may be represented as Poisson cluster processes which are age-dependent immigration-birth processes, and their existence is established. This result is used to derive some counting and interval properties of these processes using the probability generating functional.

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