Limit theorems for quadratic forms and related quantities of discretely sampled continuous-time moving averages

Department of Mathematics, Aarhus University, Ny Munkegade 118, Aarhus 8000, Denmark.

^* Corresponding author: mikkel@math.au.dk

Received: 4 July 2018
Accepted: 8 April 2019

Abstract

The limiting behavior of Toeplitz type quadratic forms of stationary processes has received much attention through decades, particularly due to its importance in statistical estimation of the spectrum. In the present paper, we study such quantities in the case where the stationary process is a discretely sampled continuous-time moving average driven by a Lévy process. We obtain sufficient conditions, in terms of the kernel of the moving average and the coefficients of the quadratic form, ensuring that the centered and adequately normalized version of the quadratic form converges weakly to a Gaussian limit.