On the central limit theorem for Toeplitz quadratic forms of stationary sequences

Let $X(t)$, $t = 0,\pm1,\ldots$, be a real-valued stationary Gaussian sequence with a spectral density function $f(\lambda)$. The paper considers the question of applicability of the central limit theorem (CLT) for a Toeplitz-type quadratic form $Q_n$ in variables $X(t)$, generated by an integrable even function $g(\lambda)$. Assuming that $f(\lambda)$ and $g(\lambda)$ are regularly varying at $\lambda=0$ of orders $\alpha$ and $\beta$, respectively, we prove the CLT for the standard normalized quadratic form $Q_n$ in a critical case $\alpha+\beta=\frac{1}{2}$.
We also show that the CLT is not valid under the single condition that the asymptotic variance of $Q_n$ is separated from zero and infinity.