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

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 LpLp-type conditions imposed on the covariance function of the model and the kernel of the quadratic functional, the process QT(t)QT(t) obeys a central limit theorem, that is, the finite-dimensional distributions of the standard T normalized process QT(t)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)QT(t), that is, the finite-dimensional distributions of the process QT(t)QT(t), normalized by TγTγ for some γ>1/2γ>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 R2R2.