zbMATH — the first resource for mathematics

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.

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60H07 Stochastic calculus of variations and the Malliavin calculus
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.