Abstract
In this paper, we obtain a large deviation principle for quadratic forms of Gaussian stationary processes. It is established by the conjunction of a result of Roch and Silbermann on the spectrum of products of Toeplitz matrices together with the analysis of large deviations carried out by Gamboa, Rouault and the first author. An alternative proof of the needed result on Toeplitz matrices, based on semi-classical analysis, is also provided.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
B. Bercu, F. Gamboa, A. Rouault, Large deviations for quadratic forms of stationary Gaussian processes. Stoch. Process. Appl. 71(1), 75–90 (1997)
A. Böttcher, B. Silbermann, Introduction to Large Truncated Toeplitz Matrices (Springer, New York, 1999)
A. Böttcher, B. Silbermann, Analysis of Toeplitz Operators, 2nd edn. Springer Monographs in Mathematics (Springer, Berlin, 2006); Prepared jointly with Alexei Karlovich
W. Bryc, A. Dembo, Large deviations for quadratic functionals of Gaussian processes. J. Theor. Probab. 10(2), 307–332 (1997); Dedicated to Murray Rosenblatt
L.A. Coburn, The C ∗ -algebra generated by an isometry. Bull. Am. Math. Soc. 73, 722–726 (1967)
A. Dembo, O. Zeitouni, Large Deviations Techniques and Applications, 2nd edn. Applications of Mathematics (New York), vol. 38 (Springer, New York, 1998)
J. Dereziński, C. Gérard, Scattering Theory of Classical and Quantum N-Particle Systems. Texts and Monographs in Physics (Springer, Berlin, 1997)
U. Grenander, G. Szegö, Toeplitz Forms and Their Applications. California Monographs in Mathematical Sciences (University of California Press, Berkeley, 1958)
T. Kato, Perturbation Theory for Linear Operators. Classics in Mathematics (Springer, Berlin, 1995); Reprint of the 1980 edition
N. Nikolski, Operators, Functions, and Systems: An Easy Reading, vol. 1, Hardy, Hankel, and Toeplitz (translated from the French by A. Hartmann), Mathematical Surveys and Monographs, vol. 92 (American Mathematical Society, RI, 2002)
M. Reed, B. Simon, Methods of Modern Mathematical Physics. I, 2nd edn. Functional Analysis (Academic, New York, 1980)
S. Roch, B. Silbermann, Limiting sets of eigenvalues and singular values of Toeplitz matrices. Asymptotic Anal. 8, 293–309 (1994)
S. Serra-Capizzano, Distribution results on the algebra generated by Toeplitz sequences: A finite-dimensional approach. Linear Algebra Appl. 328(1–3), 121–130 (2001)
H. Widom, Asymptotic behavior of block Toeplitz matrices and determinants. II. Adv. Math. 21(1), 1–29 (1976)
Acknowledgements
The authors would like to thanks A. Böttcher for providing the reference of Roch and Silbermann. They also thank the anonymous referee for his careful reading of the paper.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Bercu, B., Bony, JF., Bruneau, V. (2012). Large Deviations for Gaussian Stationary Processes and Semi-Classical Analysis. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLIV. Lecture Notes in Mathematics(), vol 2046. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27461-9_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-27461-9_19
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-27460-2
Online ISBN: 978-3-642-27461-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)