×

A noncentral limit theorem for quadratic forms of Gaussian stationary sequences. (English) Zbl 0724.60048

Summary: We examine the limit behavior of quadratic forms of stationary Gaussian sequences with long-range dependence. The matrix that characterizes the quadratic form is Toeplitz and the Fourier transform of its entries is a regularly varying function at the origin. The spectral density of the stationary sequence is also regularly varying at the origin. We show that the normalized quadratic form converges in D[0,1] to a new type of non- Gaussian self-similar process, which can be represented as a Wiener-Itô integral on \({\mathbb{R}}^ 2\).

MSC:

60G18 Self-similar stochastic processes
60G15 Gaussian processes
60G12 General second-order stochastic processes
60H05 Stochastic integrals
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Billingsley, P. (1968).Convergence of Probability Measures, Wiley, New York. · Zbl 0172.21201
[2] Bingham, N., Goldie, C. M., and Teugels, J. L. (1987).Regular Variation, Cambridge University Press, Cambridge, England. · Zbl 0617.26001
[3] Chow, Y. S., and Teicher, H. (1978).Probability Theory, Springer-Verlag, New York. · Zbl 0399.60001
[4] Dobrushin, R. L., and Major, P. (1979). Non-central limit theorems for non-linear functions of Gaussian fields.Z. Wahr. verw. Gebiete 50, 27-52. · Zbl 0397.60034 · doi:10.1007/BF00535673
[5] Fox, R., and Taqqu, M. S. (1987). Central limit theorems for quadratic forms in random variables having long range dependence.Probab. Theor. Rel. Fields 74, 213-240. · Zbl 0586.60019 · doi:10.1007/BF00569990
[6] Major, P. (1980).Multiple Wiener-Itô Integrals, Lecture Notes in Mathematics, Vol. 849, Springer-Verlag, New York.
[7] Rosenblatt, M. (1961). Independence and dependence.Proc. 4th Berkeley Symp. Math. Statist. Probab. 2, 431-443. · Zbl 0105.11802
[8] Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and the Resenblatt process.Z. Wahr. verw. Gebiete 31, 287-302. · Zbl 0303.60033 · doi:10.1007/BF00532868
[9] Terrin, N., and Taqqu, M. S. Power counting theorem onR n . (To be published.)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.