# What is the autocorrelation of a squared Gaussian process?

Suppose ${X_t; t \in R}$ is a wss, zero mean Gaussian random process with autocorrelation function $R_X( \tau) ; \tau \in R$ and power spectral density $S_X(\omega); \omega \in R$. If w define the random process ${Y_t;t\in R}$ by ${Y_t = ({X_t}^2)}$

What is the autocorrelation function of $Y_t$ (in terms of $\tau$)?

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If $X_t$ and $X_{t+\tau}$ are jointly Gaussian with means $0$, variances $\sigma^2$ and covariance $c$, then $\text{Cov}(X_t^2, X_{t+\tau}^2) = 2 c^2$. One way to get this is from Isserlis's theorem.