Gaussian limit for determinantal random point fields. (English) Zbl 1033.60063

A random point field on a locally compact Hausdorff space \(E\) with a measure \(\mu\) is considered. The appearance of \(n\) points of the field at \(n\) disjoint subsets of \(E\) is proposed to be determined by the \(n\)-dimensional density \(p_n\) with respect to \(\mu^n\). The author assumes that there exists an integral operator with a kernel \(K\) such that for any \(n\geq 2\) the density \(p_n\) is representable in the form of determinant of the kernel \(K\). Three theorems are proved. The first one in the case of general \(E\) gives sufficient conditions for the sequence of centered and normed linear functionals of the point field to converge in distribution to a Gaussian random value. The second one for \(E= R^d\) \((d\geq 2)\) and “almost” homogeneous kernel \(K\) gives conditions for the weak convergence of the sequence of the rescaled point fields to the Gaussian white noise. And the third one for \(E=R^1\) and homogeneous kernel \(K\) asserts the weak convergence of a sequence of rescaled point processes to a self-similar Gaussian process with the spectral density \(| k|^\alpha\) \((0< \alpha< 1)\).


60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G18 Self-similar stochastic processes
Full Text: DOI arXiv


[1] BASOR, E. (1997). Distribution functions for random variables for ensembles of positive Hermitian matrices. Comm. Math. Phys. 188 327-350. · Zbl 0905.47016
[2] BASOR, E. and WIDOM, H. (1999). Determinants of Airy operators and applications to random matrices. J. Statist. Phys. 96 1-20. · Zbl 0964.82023
[3] COSTIN, O. and LEBOWITZ, J. (1995). Gaussian fluctuations in random matrices. Phys. Rev. Lett. 75 69-72.
[4] DALEY, D. J. and VERE-JONES, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York. · Zbl 0657.60069
[5] DEIFT, P. (1999). Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Amer. Math Soc., Alexandria, VA. · Zbl 0997.47033
[6] DIACONIS, P. and EVANS, S. N. (2000). Immanants and finite point processes. J. Combin. Theory Ser. A 91 305-321. · Zbl 0965.15007
[7] DIACONIS, P. and EVANS, S. N. (2001). Linear functionals of eigenvalues of random matrices. Trans. Amer. Math. Soc. 353 2615-2633. JSTOR: · Zbl 1008.15013
[8] DIACONIS, P. and SHAHSHAHANI, M. (1994). On the eigenvalues of random matrices. J. Appl. Probab. 31A (Special vol.) 49-62. JSTOR: · Zbl 0807.15015
[9] DOBRUSHIN, R. L. (1978). Automodel generalized random fields and their renorm-group. In Multicomponent Random Systems (R. L. Dobrushin and Ya. G. Sinai, eds.). Dekker, New York. · Zbl 0499.60047
[10] DOBRUSHIN, R. L. (1979). Gaussian and their subordinated self-similar random generalized fields. Ann. Probab. 7 1-28. · Zbl 0392.60039
[11] GELFAND, I. M. and VILENKIN, N. Ya. (1964). Generalized Functions IV: Some Applications of Harmonic Anaysis. Academic Press, New York.
[12] HIDA, T. (1980). Brownian Motion. Springer, New York. · Zbl 0423.60063
[13] JOHANSSON, K. (1998). On fluctuation of eigenvalues of random Hermitian matrices. Duke Math. J. 91 151-204. · Zbl 1039.82504
[14] JOHANSSON, K. On random matrices from the compact classical groups. Ann. Math. 145 519- 545 JSTOR: · Zbl 0883.60010
[15] KARAMATA, J. (1930). Sur un mode de croissance régulière des fonctions. Mathematica (Cluj) 4 38-53. · JFM 56.0907.01
[16] KARAMATA, J. (1933). Sur un mode de croissance régulière. Théorèmes fondamentaux. Bull. Soc. Math. France 61 55-62. · Zbl 0008.00807
[17] LENARD, A. (1973). Correlation functions and the uniqueness of the state in classical statistical mechanics. Comm. Math. Phys. 30 35-44.
[18] LENARD, A. (1975). States of classical statistical mechanical system of infinitely many particles I. Arch. Rational Mech. Anal. 59 219-239.
[19] LENARD, A. (1975). States of classical statistical mechanical systems of infinitely many particles II. Arch. Rational Mech. Anal. 59 240-256.
[20] LUKACS, E. (1970). Characteristic Functions, 2nd. ed. Griffin, London. · Zbl 0198.23804
[21] MACCHI, O. (1975). The coincidence approach to stochastic point processes. Adv. Appl. Probab. 7 82-122. JSTOR: · Zbl 0366.60081
[22] MACCHI, O. (1977). The fermion process-a model of stochastic point process with repulsive points. In Transactions of the Seventh Prague Conference on Information Theory, Statistical Decision Functions, Random Processes and of the Eighth European Meeting of Statisticians A 391-398. Reidel, Dordrecht. · Zbl 0413.60048
[23] REED, M. and SIMON, B. (1975-1980). Methods of Modern Mathematical Physics 1-4. Academic Press, New York.
[24] SENETA, E. (1976). Regularly Varying Functions. Lecture Notes in Math. 508. Springer, New York. · Zbl 0324.26002
[25] SINAI, Ya. (1976). Automodel probability distributions. Theory Probab. Appl. 21 273-320.
[26] SOSHNIKOV, A. (2000). Determinantal random point fields. Russian Math. Surveys 55 923- 975. Available via xxx.lanl.gov/abs/math/0002099. · Zbl 0991.60038
[27] SOSHNIKOV, A. (2000). Central limit theorem for local linear statistics in classical compact groups and related combinatorial identities. Ann. Probab. 28 1353-1370. · Zbl 1021.60018
[28] SOSHNIKOV, A. (2000). Gaussian fluctuations in Airy, Bessel, sine and other determinantal random point fields. J. Statist. Phys. 100 491-522. · Zbl 1041.82001
[29] SPOHN, H. (1987). Interacting Brownian particles: A study of Dyson’s model. In Hydrodynamic Behavior and Interacting Particle Systems (G. Papanicolau, ed.). Springer, New York. · Zbl 0674.60096
[30] SHIRAI, T. and TAKAHASHI, Y. (2000). Random point fields associated with certain Fredholm determinants II: Fermion shifts and their ergodic and Gibbs properties. · Zbl 1051.60053
[31] WIEAND, K. (1998). Eigenvalue distribution of random matrices in the permutation group and compact Lie groups. Ph. D. dissertation, Dept. Mathematics, Harvard Univ.
[32] DAVIS, CALIFORNIA 95616 E-MAIL: soshniko@math.ucdavis.edu
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.