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Extra resources for An Introduction to the Theory of Point Processes
After the war, the characteristic functional was discussed by Le Cam (1947) for stochastic processes and Bochner (1947) for random interval functions. Bochner's (1955) monograph, in particular, contains many original ideas that have only partially been followed up, for example, by Brillinger (1972). Kendall (1949) and Bartlett and Kendall (1951) appear to be the first to have used the characteristic functional in the study of specific population models. ) defined by G[h] = E [I) J h(x;) = E[eflogh(x)N(dx>], where h(x) is a suitable test function and the X; are the points at which population members are located, that is, the atoms of the counting measures N( · ).
Boundedly finite and without fixed atoms. V. 0 To extend this result to the nonorderly case, consider for fixed real z ::5; 1, the set function z in ::5; Qz(A) =: -log E(zN) =: -log Pz(A) defined over the Borel sets A. VI it follows that Qz( ·)is a measure, absolutely continuous with respect to Q( · ). 3) Qz(A) L = qz(x)Q(dx) and for Q-almost all x [cf. 2) for qz(x) s•. 1 for a discussion of this property of Radon-Nikodym derivatives). If we continue to assume that the process has no fixed atoms, Q(S,,xl and hence also Qz(S,,x) both-+ 0 as e-+ 0, for then s•.
Starting in the early 1960s, Matthes and co-workers developed many aspects concerned with infinitely divisible point processes and related questions. The book by Kerstan, Matthes, and Mecke (1974) represented the culmination of the first decade of such work; extensive revisions and new material were incorporated into the later editions in English (1978) (referred to as MKM in this book) and in Russian (1982). In applications, these ideas have been useful not only in queueing theory [for continuing development in this field see the monographs of Franken et al.