×

zbMATH — the first resource for mathematics

Poisson process Fock space representation, chaos expansion and covariance inequalities. (English) Zbl 1233.60026
Let \(\eta\) be a Poisson process on a measurable space \((\mathbb Y, \mathcal Y)\) with \(\sigma\)-finite intensity measure \(\lambda\). In contrast to the literature, the authors do not make any restrictions of generality, they neither impose a topological structure on \((\mathbb Y, \mathcal Y)\), nor assume the measure \(\lambda\) to be continuous.
The paper is organized as follows: Section 2 establishes an explicit Fock space representation of square integrable functions of \(\eta\). In Section 3, the authors identify explicitly, in terms of iterated difference operators, the integrand in the Wiener-Itô chaos expansion. Section 4 applies the results of the previous two sections to extend well-known variance inequalities for homogeneous Poisson processes on the line to the general Poisson case. In Section 5, the authors derive covariance identities for Poisson processes on (strictly) ordered spaces and Harris-FKG-inequalities for monotone functions of \(\eta\). In Section 6, the authors discuss the Poincaré and Harris-FKG inequalities for infinitely divisible random measures. Section 7 describes some of the implications in the case where \(\mathbb Y\) is a finite set.

MSC:
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G51 Processes with independent increments; Lévy processes
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Barbour A.D., Holst L., Janson S.: Poisson Approximation. Clarendon Press, Oxford (1992)
[2] Blaszczyszyn B.: Factorial-moment expansion for stochastic systems. Stoch. Proc. Appl. 56, 321–335 (1995) · Zbl 0816.60042 · doi:10.1016/0304-4149(94)00071-Z
[3] Bollobás B., Riordan O.: Percolation. Cambridge University Press, New York (2006)
[4] Chen L.: Poincaré-type inequalities via stochastic integrals. Z. Wahrscheinlichkeitstheorie verw. Gebiete 69, 251–277 (1985) · Zbl 0549.60019 · doi:10.1007/BF02450283
[5] Daley D.J., Vere-Jones D.: An Introduction to the Theory of Point Processes, vol. II, 2nd edn. Springer, New York (2008) · Zbl 1159.60003
[6] Dellacherie, C., Meyer, P.A.: Probabilities and potential. In: Mathematics Studies, vol. 29. Hermann, Paris; North-Holland, Amsterdam (1978) · Zbl 0494.60001
[7] Efron B., Stein C.: The jackknife estimate of variance. Ann. Stat. 9, 586–596 (1981) · Zbl 0481.62035 · doi:10.1214/aos/1176345462
[8] Georgii H., Küneth T.: Stochastic order of point processes. J. Appl. Prob. 34, 868–881 (1997) · Zbl 0905.60030 · doi:10.2307/3215003
[9] Heveling M., Reitzner M.: Poisson-Voronoi approximation. Ann. Appl. Probab. 19, 719–736 (2009) · Zbl 1172.60003 · doi:10.1214/08-AAP561
[10] Hitsuda, M.: Formula for Brownian partial derivatives. In: Proceedings of the 2nd Japan-USSR Symposium on Probability Theory, pp. 111–114 (1972)
[11] Houdré C., Perez-Abreu V.: Covariance identities and inequalities for functionals on Wiener space and Poisson space. Ann. Probab. 23, 400–419 (1995) · Zbl 0831.60029 · doi:10.1214/aop/1176988392
[12] Houdré C.: Remarks on deviation inequalities for functions of infinitely divisible random vectors. Ann. Probab. 30, 1223–1237 (2002) · Zbl 1017.60018 · doi:10.1214/aop/1029867126
[13] Houdré C., Privault N.: Concentration and deviation inequalities in infinite dimensions via covariance representations. Bernoulli 8, 697–720 (2002) · Zbl 1012.60020
[14] Itô K.: Multiple Wiener integral. J. Math. Soc. Jpn. 3, 157–169 (1951) · Zbl 0044.12202 · doi:10.2969/jmsj/00310157
[15] Itô K.: Spectral type of the shift transformation of differential processes with stationary increments. Trans. Am. Math. Soc. 81, 253–263 (1956) · Zbl 0073.35303 · doi:10.2307/1992916
[16] Ito Y.: Generalized Poisson functionals. Probab. Theory Relat. Fields 77, 1–28 (1988) · Zbl 0617.60035 · doi:10.1007/BF01848128
[17] Kabanov Y.M.: On extended stochastic integrals. Theory Probab. Appl. 20, 710–722 (1975) · Zbl 0355.60047 · doi:10.1137/1120080
[18] Kabanov, Y.M., Skorokhod, A.V.: Extended stochastic integrals. In: Proceedings of the School-Seminar on the Theory of Random Processes, Part I. Druskininkai, November 25–30, 1974. Vilnius (Russian) (1975)
[19] Kallenberg O.: Random Measures. Akademie-Verlag, Berlin; Academic Press, London (1983) · Zbl 0544.60053
[20] Kallenberg O.: Foundations of Modern Probability, 2nd edn. Springer, New York (2002) · Zbl 0996.60001
[21] Last, G., Penrose, M.D.: Martingale representation for Poisson processes with applications to minimal variance hedging. arXiv 1001.3972 (2010)
[22] Liebscher V.: On the isomorphism of Poisson space and symmetric Fock Space. In: Accardi, L. (eds) Quantum Probability and Related Topics IX, pp. 295–300. World Scientific, Singapore (1994)
[23] Løkka A.: Martingale representation of functionals of Lévy processes. Stoch. Anal. Appl. 22, 867–892 (2005) · Zbl 1060.60058 · doi:10.1081/SAP-120037622
[24] Matthes K., Kerstan J., Mecke J.: Infinitely Divisible Point Processes. Wiley, Chichester (1978) · Zbl 0383.60001
[25] Mecke J.: Stationäre zufällige Maße auf lokalkompakten Abelschen Gruppen. Z. Wahrsch. verw. Gebiete 9, 36–58 (1967) · Zbl 0164.46601 · doi:10.1007/BF00535466
[26] Meester, R., Roy, R.: Continuum Percolation. Cambridge University Press (1996) · Zbl 0858.60092
[27] Meyer P.A.: Quantum Probability for Probabilists, 2nd edn. Springer, Berlin (1995) · Zbl 0877.60079
[28] Møller J., Zuyev S.: Gamma-type results and other related properties of Poisson processes. Adv. Appl. Probab. 28, 662–673 (1996) · Zbl 0857.60008 · doi:10.2307/1428175
[29] Molchanov I., Zuyev S.: Variational analysis of functionals of Poisson processes. Math. Oper. Res. 25, 485–508 (2000) · Zbl 1018.49022 · doi:10.1287/moor.25.3.485.12217
[30] Nualart, D., Vives, J.: Anticipative calculus for the Poisson process based on the Fock space. In: Séminaire de Probabilités XXIV. Lecture Notes in Mathematics, vol. 1426, pp. 154–165 (1990)
[31] Ogura H.: Orthogonal functionals of the Poisson processes. Trans. IEEE Inf. Theory 18, 473–481 (1972) · Zbl 0244.60044 · doi:10.1109/TIT.1972.1054856
[32] Peccati G., Solé J.L., Taqqu M.S., Utzet F.: Stein’s method and normal approximation of Poisson functionals. Ann. Probab. 38, 443–478 (2010) · Zbl 1195.60037 · doi:10.1214/09-AOP477
[33] Penrose M.: Random Geometric Graphs. Oxford University Press, Oxford (2003) · Zbl 1029.60007
[34] Penrose M.D., Sudbury A.: Exact and approximate results for deposition and annihilation processes on graphs. Ann. Appl. Probab. 15, 853–889 (2005) · Zbl 1073.60097 · doi:10.1214/105051604000000765
[35] Penrose M.D., Wade A.R.: Multivariate normal approximation in geometric probability. J. Stat. Theory Pract. 2, 293–326 (2008) · doi:10.1080/15598608.2008.10411876
[36] Picard J.: On the existence of smooth densities for jump processes. Probab. Theory Relat. Fields 105, 481–511 (1996) · Zbl 0853.60064 · doi:10.1007/BF01191910
[37] Privault N.: Extended covariance identities and inequalities. Stat. Probab. Lett. 55, 247–255 (2001) · Zbl 1078.60032 · doi:10.1016/S0167-7152(01)00140-7
[38] Surgailis D.: On multiple Poisson stochastic integrals and associated Markov semigroups. Probab. Math. Stat. 3, 217–239 (1984) · Zbl 0548.60058
[39] Skorohod A.V.: On a generalization of a stochastic integral. Theory Probab. Appl. 20, 219–233 (1975) · Zbl 0333.60060 · doi:10.1137/1120030
[40] Wiener N.: The homogeneous chaos. Am. J. Math. 60, 897–936 (1938) · Zbl 0019.35406 · doi:10.2307/2371268
[41] Wu, L.: L 1 and modified logarithmic Sobolev inequalities and deviation inequalities for Poisson point processes. Preprint (1998)
[42] Wu L.: A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Relat. Fields 118, 427–438 (2000) · Zbl 04564032 · doi:10.1007/PL00008749
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.