Generalized Szegő theorems and asymptotics of cumulants by graphical methods
HTML articles powered by AMS MathViewer
- by Florin Avram PDF
- Trans. Amer. Math. Soc. 330 (1992), 637-649 Request permission
Abstract:
We obtain some general asymptotics results about a class of deterministic sums called "sums with dependent indices," which generalize a classical theorem of Szegö. The above type of sums is encountered when establishing convergence to the Gaussian distribution of sums of Wick products by the method of cumulants. Our asymptotic results reduce in this situation the proof of the central limit theorem to the study of the connectivity of a family of associated graphs.References
- Florin Avram and Lawrence Brown, A generalized Hölder inequality and a generalized Szegő theorem, Proc. Amer. Math. Soc. 107 (1989), no. 3, 687–695. MR 984781, DOI 10.1090/S0002-9939-1989-0984781-X
- F. Avram, On bilinear forms in Gaussian random variables and Toeplitz matrices, Probab. Theory Related Fields 79 (1988), no. 1, 37–45. MR 952991, DOI 10.1007/BF00319101
- Florin Avram and Robert Fox, Central limit theorems for sums of Wick products of stationary sequences, Trans. Amer. Math. Soc. 330 (1992), no. 2, 651–663. MR 1034656, DOI 10.1090/S0002-9947-1992-1034656-X R. Bixby, Matroids and operations research, Preprint.
- Peter Breuer and Péter Major, Central limit theorems for nonlinear functionals of Gaussian fields, J. Multivariate Anal. 13 (1983), no. 3, 425–441. MR 716933, DOI 10.1016/0047-259X(83)90019-2
- Victor Bryant and Hazel Perfect, Independence theory in combinatorics, Chapman and Hall Mathematics Series, Chapman & Hall, London-New York, 1980. An introductory account with applications to graphs and transversals. MR 604173, DOI 10.1007/978-94-009-5900-2
- Robert Fox and Murad S. Taqqu, Noncentral limit theorems for quadratic forms in random variables having long-range dependence, Ann. Probab. 13 (1985), no. 2, 428–446. MR 781415
- Robert Fox and Murad S. Taqqu, Central limit theorems for quadratic forms in random variables having long-range dependence, Probab. Theory Related Fields 74 (1987), no. 2, 213–240. MR 871252, DOI 10.1007/BF00569990
- Ulf Grenander and Gabor Szegö, Toeplitz forms and their applications, California Monographs in Mathematical Sciences, University of California Press, Berkeley-Los Angeles, 1958. MR 0094840
- L. Giraitis and D. Surgailis, CLT and other limit theorems for functionals of Gaussian processes, Z. Wahrsch. Verw. Gebiete 70 (1985), no. 2, 191–212. MR 799146, DOI 10.1007/BF02451428 —, Multivariate Appell polynomials and the central limit theorem, Dependence and Independence in Probability and Statistics, (E.. Eberlein and M. Taqqu, eds.), Birkhäuser, 1986.
- Yitzhak Katznelson, An introduction to harmonic analysis, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0248482
- D. J. A. Welsh, Matroid theory, L. M. S. Monographs, No. 8, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0427112
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 330 (1992), 637-649
- MSC: Primary 60F05
- DOI: https://doi.org/10.1090/S0002-9947-1992-1059708-X
- MathSciNet review: 1059708