Limit theorems for quadratic forms of Lévy-driven continuous-time linear processes

• Published in: Stochastic processes and their applications 2016-04, Vol.126 (4), p.1036-1065
• Main Authors: Bai, Shuyang, Ginovyan, Mamikon S., Taqqu, Murad S.
• Format: Article
• Language: English
• Subjects: Brownian motion; Central and non-central limit theorems; Long memory; Lévy process; Teugels martingale; Toeplitz type quadratic functional; Wiener–Itô integral
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/j.spa.2015.10.010

We study the asymptotic behavior of a suitable normalized stochastic process {QT(t),t∈[0,1]}. This stochastic process is generated by a Toeplitz type quadratic functional of a Lévy-driven continuous-time linear process. We show that under some Lp-type conditions imposed on the covariance function of the model and the kernel of the quadratic functional, the process QT(t) obeys a central limit theorem, that is, the finite-dimensional distributions of the standard T normalized process QT(t) tend to those of a normalized standard Brownian motion. In contrast, when the covariance function of the model and the kernel of the quadratic functional have a slow power decay, then we have a non-central limit theorem for QT(t), that is, the finite-dimensional distributions of the process QT(t), normalized by Tγ for some γ>1/2, tend to those of a non-Gaussian non-stationary-increment self-similar process which can be represented by a double stochastic Wiener–Itô integral on R2.