  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.
