Stochastic analysis of multivariate point processes

Abstract

This thesis is concerned with multivariate point processes in R¹. For the purposes of this general survey a multivariate point process may be thought of as a series of events of finitely many distinguishable types happening in time. Chapter One defines a multivariate point process and shows that such a process is uniquely specified once a consistent set of finite-dimensional distributions is given. This result is essentially known. Notions such as those of independence, superposition, moment measures, stationarity, intensities, parameters, orderliness, fixed atoms, convergence in distribution, triangular array, and complete randomness are then defined. Most of these are fairly straightforward extensions of the definitions for univariate point processes. Finally, we present some examples of multivariate point processes and define, in particular, what we mean by a Poisson process. The next chapter is based on ideas in Milne ( 1971) but the results are presented here for bivariate point processes in R¹ instead of for univariate point processes in Rⁿ as in that paper. The basic result, well-known in the univariate case, is an extension to non-orderly processes of Korolyuk's theorem connecting the intensity and the parameter. We give related results for higher-order moments and some stationarity results which are used later. Our methods are extensions of techniques of Leadbetter (1968) and are capable of further extension e.g. to processes in Rⁿ (Milne, 1971). In Chapter Three we study extensions to multivariate point processes of the Palm functions introduced for univariate point processes by Palm (1943) and Khinchin (1955). These functions are of interest in their own right as well as being useful in later discussion of superposition results. It is shown that the usual subadditivity and convexity methods appear to be inadequate for a full treatment of Palm functions in the multivariate case but that we can proceed using extensions of recent techniques of Belyaev (1968, 1970) and Leadbetter (1970). Next, we derive some generalizations of the sc-called Palm-Khinchin formulae for univariate point processes. Examples of bivariate Palm functions are exhibited for the randomly translated Poisson process and given an intuitive interpretation. Finally, the representation due to Fieger (196h) for the probabilities of a general non-orderly, stationary, univariate point process is derived from an extension of our representation in terms of multivariate Palm functions for the probabilities of a stationary, strongly orderly, multivariate point process. The fourth chapter introduces probability generating functionals for multivariate point processes. These are our main tool in later discussion of infinite divisibility and superpositions. Most of the results are extensions of previous work for univariate point processes (Moyal, 1962; Vere-Jones, 1968, 1970; Westcott, 19715) but we pay special attention to the complications arising from fixed atoms. An example is given to illustrate the complications which arise from such fixed atoms when convergence in distribution is discussed. The concept of independence for multivariate point processes is considered in relation to the probability generating functional. Infinitely divisible multivariate point processes are introduced in Chapter Five which outlines some results about their finite-dimensional distributions and gives a constructive derivation of the canonical form of the probability generating functional of such a process. Multivariate Poisson cluster processes are considered and the randomly translated Poisson process looked at from this point of view. We then investigate more general infinitely divisible bivariate Poisson processes, answer some questions raised by Cox and Lewis (1970), and make connections with recent work of Newman (1970), and Milne and Westcott (1972) on Gauss- Poisson processes. Finally, some results on convergence of infinitely divisible multivariate point processes are derived. This last result is applied in Chapter Six in discussing the convergence of the 'row sums’ of a triangular array of multivariate point processes to a multivariate Poisson process. We first consider convergence to a general infinitely divisible multivariate process and then specialize our result to the case of convergence to an infinitely divisible multivariate Poisson process with independent marginals. Also, in this case the conditions for convergence are rephrased in terms of multivariate Palm functions using the results of Chapter Three and connections made with the previous work by Khinchin (1955), Ososkov (1956), Grigelionis (1963) on univariate point processes and by Cinlar (l968) on multivariate point processes. It is shown that a superposition theorem of Vere-Jones (1968) is an interesting special case of the result of Grigelionis (1963) and hence that the conditions of the former theorem may be made necessary as well as sufficient. Lastly, as a diversion to illustrate a direct approach to superposition problems, we improve slightly a theorem of Goldman (l967b) about convergence to a stationary univariate Poisson process in Rⁿ . The final chapter returns to the oft-recurring randomly translated Poisson process to discuss a special identifiability problem viz, how much information a complete input-output record contains about the displacement distribution. The result for Poisson processes in R¹ is contained in Milne (1970) and this chapter shows how, with minor modifications to the argument, the result may be extended to Poisson processes in Rⁿ i.e. we consider a Poisson process in Rn randomly displaced by a bivariate distribution. It is shown that, from a complete input-output record, the displacement distribution is identifiable with probability one. This result flows essentially from an application of the pointwise ergodic theorem using some results which are derived about the form of some joint distributions. The connection of this identifiability result with recent work of Brown (1970) is also discussed.