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Moment formulae for general point processes. (English) Zbl 1297.60031
Summary: The goal of this paper is to generalize most of the moment formulae obtained in [N. Privault, Ann. Inst. Henri Poincaré, Probab. Stat. 48, No. 4, 947–972 (2012; Zbl 1278.60084)]. More precisely, we consider a general point process $$\mu$$, and show that the quantities relevant to our problem are the so-called Papangelou intensities. When the Papangelou intensities of $$\mu$$ are well-defined, we show some general formulae to recover the moment of order $$n$$ of the stochastic integral of the point process. We will use these extended results to introduce a divergence operator and study a random transformation of the point process.

##### MSC:
 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60H07 Stochastic calculus of variations and the Malliavin calculus
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##### References:
 [1] Camilier, I.; Decreusefond, L., Quasi-invariance and integration by parts for determinantal and permanental processes, J. Funct. Anal., 259, 1, 268-300, (2010) · Zbl 1203.60050 [2] Daley, D. J.; Vere-Jones, D., An introduction to the theory of point processes. vol. I. elementary theory and methods, Probab. Appl. (N. Y.), (2003), Springer-Verlag New York · Zbl 1026.60061 [3] Daley, D. J.; Vere-Jones, D., An introduction to the theory of point processes. vol. II. general theory and structure, Probab. Appl. (N. Y.), (2008), Springer New York · Zbl 1159.60003 [4] Georgii, H.; Yoo, H. J., Conditional intensity and Gibbsianness of determinantal point processes, J. Stat. Phys., 118, 1-2, 55-84, (2005) · Zbl 1130.82016 [5] Hough, J. B.; Krishnapur, M.; Peres, Y.; Virág, B., Determinantal processes and independence, Probab. Surv., 3, 206-229, (2006), (electronic) · Zbl 1189.60101 [6] Kallenberg, O., Random measures, (1986), Akademie-Verlag Berlin · Zbl 0288.60053 [7] Lenard, A., States of classical statistical mechanical systems of infinitely many particles. I, Arch. Ration. Mech. Anal., 59, 3, 219-239, (1975) [8] Lenard, A., States of classical statistical mechanical systems of infinitely many particles. II. characterization of correlation measures, Arch. Ration. Mech. Anal., 59, 3, 241-256, (1975) [9] Mecke, J., Stationäre zufällige masse auf lokalkompakten abelschen gruppen, Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 9, 36-58, (1967) · Zbl 0164.46601 [10] Nguyen, X.; Zessin, H., Integral and differential characterizations of the Gibbs process, Math. Nachr., 88, 105-115, (1979) · Zbl 0444.60040 [11] Privault, N., Stochastic analysis in discrete and continuous settings with normal martingales, Lecture Notes in Math., vol. 1982, (2009), Springer-Verlag Berlin · Zbl 1185.60005 [12] Privault, N., Invariance of Poisson measures under random transformations, Ann. Inst. H. Poincaré, 48, 947-972, (2012) · Zbl 1278.60084 [13] Privault, N., Moments of Poisson stochastic integrals with random integrands, Probab. Math. Statist., 32, 2, 227-239, (2012) · Zbl 1268.60069 [14] Ruelle, D., Statistical mechanics: rigorous results, (1969), W.A. Benjamin, Inc. New York, Amsterdam · Zbl 0177.57301 [15] Ruelle, D., Superstable interactions in classical statistical mechanics, Comm. Math. Phys., 18, 127-159, (1970) · Zbl 0198.31101 [16] Shirai, T.; Takahashi, Y., Random point fields associated with certain Fredholm determinants. II. fermion shifts and their ergodic and Gibbs properties, Ann. Probab., 31, 3, 1533-1564, (2003) · Zbl 1051.60053 [17] Soshnikov, A., Determinantal random point fields, Uspekhi Mat. Nauk, 55, 5(335), 107-160, (2000) · Zbl 0991.60038 [18] Torrisi, G. L., Point processes with papangelou conditional intensity: from the Skorohod integral to the Dirichlet form, Markov Process. Related Fields, 19, 195-248, (2013) · Zbl 1308.58021 [19] Yoo, H. J., Gibbsianness of fermion random point fields, Math. Z., 252, 1, 27-48, (2006) · Zbl 1081.60070
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