= Covariance of quadratic transforms of a linear process
For $X_t=\sum_{j\geq0}\psi_j\eta_{t-j}$ driven by centered unit-variance iid noise with a finite fourth moment, put $\kappa_4=E\eta^4-3$ and $m_3=E\eta^3$. Independence gives
$$
\operatorname{Cov}(X_t^2,X_{t-h}^2)=2\gamma(h)^2+\kappa_4\sum_{j\geq0}\psi_j^2\psi_{j+|h|}^2.
$$
The sum of the two linear-quadratic cross covariances is $m_3\sum_{j\geq0}(\psi_j^2\psi_{j+|h|}+\psi_j\psi_{j+|h|}^2)$. The <cumulant> term disappears for Gaussian noise. Merely assuming <strong white noise> does not justify the Gaussian formula; a fourth moment is needed for the variance of the quadratic transform.
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