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

TL;DR

This paper investigates the asymptotic distribution of quadratic forms of discretely sampled continuous-time moving averages driven by Lévy processes, establishing conditions for Gaussian convergence.

Contribution

It provides new sufficient conditions for the weak convergence of quadratic forms of sampled continuous-time moving averages to a Gaussian distribution.

Findings

Quadratic forms converge to Gaussian limits under specified conditions.

Conditions depend on the kernel of the moving average and quadratic form coefficients.

Results extend understanding of spectral estimation for continuous-time processes.

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 L\'{e}vy process. We obtain sufficient conditions, in terms of the kernel of the moving 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.