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Factorial moment espansion for stochastic systems. (English) Zbl 0816.60042
Summary: For a given functional of a simple point process, we find an analogue of Taylor’s theorem for its mean value. The terms of the expansion are integrals of some real functions with respect to factorial moment measures of the point process. The remainder term is an integral of some functional with respect to a higher order Campbell measure. A special case of this expansion is Palm-Khinchin formula. The results complement previous studies of M. I. Reiman and B. Simon [Math. Oper. Res. 14, No. 1, 26-59 (1989; Zbl 0665.60105)], F. Baccelli and P. Brémaud [Adv. Appl. Probab. 25, No. 1, 221-224 (1993; Zbl 0787.60057)] and shed new light on light traffic approximations of D. J. Daley and T. Rolski [Stochastic Processes Appl. 49, No. 1, 141- 158 (1994; Zbl 0788.60113)] and the author and T. Rolski [Ann. Appl. Probab. 3, No. 3, 881-896 (1993; Zbl 0781.60081)].

MSC:
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60K25 Queueing theory (aspects of probability theory)
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