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By Paruchuri R Krishnaiah

ISBN-10: 0124266010

ISBN-13: 9780124266018

ISBN-10: 0124266029

ISBN-13: 9780124266025

ISBN-10: 0124266037

ISBN-13: 9780124266032

ISBN-10: 0124266045

ISBN-13: 9780124266049

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2. 3. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 1. 2. 3. 4. 5. 6. General Parameter Estimation for Point Processes References 34 34 36 38 39 39 44 52 55 61 68 73 79 84 90 94 112 112 115 117 119 121 125 128 t This research was partially supported by the J. S. Guggenheim Memorial Foundation and National Science Foundation Grant MCS76-06117. 33 Copyright © 1978 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-4266 01-0 34 1. David R. Brillinger INTRODUCTION Both the literature concerning the subject of ordinary time series analysis and that of point process analysis are large and are rapidly growing.

Stable processes are usually defined through the characteristic functional (see Bochner, 1960). As a final means of introducing stochastic time series, consider the class of harmonizable processes. , as a limit in the mean of approximat­ ing Stieltjes sums). Harmonizable processes are discussed by Loeve (1955, p. 474). X(t, ω) is seen to be a linear function of Z(A, ω) so its moments can be given directly in terms of those of Z(A, ω). This manner of introducing a time series proves especially useful in the case of stationary series.

This is the way the Gauss-Poisson process is defined, for example (see Milne and Westcott, 1972), as well as the infinitely divisible point process. , see Vere-Jones, 1968). 3) may be determined from G[£] by setting ξ(ί) = zk = 1 for t in Ik otherwise and then determining the coefficient of f[l · · · z"KK in the Taylor series expan­ sion of the functional. An approach to the foundations of point processes which is related to those just given, but yet has a somewhat different character, involves taking Ω to be the set of all locally finite subsets, ω, of [0, oo).

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