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The Extended Probability Generating Functional, with Application to Mixing Properties of Cluster Point Processes
Author(s) -
Daley D. J.,
VereJones D.
Publication year - 1987
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19871310125
Subject(s) - mathematics , pointwise , bounded function , mixing (physics) , separable space , combinatorics , metric space , measurable function , space (punctuation) , cluster (spacecraft) , point process , mathematical analysis , physics , statistics , linguistics , philosophy , quantum mechanics , computer science , programming language
An extended probability generating functional (p. g. fl.) \documentclass{article}\pagestyle{empty}\begin{document}$ \bar{G}[h]\; = \;E\;\left({\exp \;\int\limits_x {\log } \;h(x)\; \times \;N(dx)} \right) $\end{document} is well‐defined for any point process N on the complete separable metric space χ over the space V̄ 0 of measurable functions h : χ → (0, 1) such that inf x cH h ( x ) > 0. The distribution of N is determined uniquely by the p.g.fl. G [ h ] ≡ Ḡ[ h ] over the smaller space V 0 of functions h ε V̄ 0 for which 1 — h has bounded support. Continuity results for Ḡ[·] involving pointwise convergent sequences { h n } V 0 or V̄ 0 or V̄ ≡ {measurable h : χ → [0, 1]} or V = { h ε V̄: 1 — h has bounded support} are reviewed, and used in furnishing a complete p. g. fl. proof of the mixing property of certain stationary cluster processes.