Infinite moving-average representation ($X_t-\mu ={\sum }_{j\ge 0}{\psi }_j{\epsilon }_{t-j}$)

A causal square-integrable linear time series can be written $X_t=\mu +{\sum }_{j\ge 0}{\psi }_j{\epsilon }_{t-j}$. Square summability of the coefficients ensures mean-square convergence for white noise. An innovation representation additionally identifies its driving noise as the innovation process.

For $X_t={\sum }_{j\ge 0}{\psi }_j{\eta }_{t-j}$ 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 givesThe sum of the two linear-quadratic cross covariances is $m_3{\sum }_{j\ge 0}({\psi }_j^2{\psi }_{j+|h|}+{\psi }_j{\psi }_{j+|h|}^2)$. 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.