Consider the stochastic process $a(t) \in \mathbb{C}$. Its autocorrelation function is given as $$ \phi_{aa}(\tau)=\left(a(t)\star a(t)\right)(\tau)=\int_{-\infty}^{\infty}a^*(t)\cdot a(t+\tau) \mathrm{d}t $$ I now want to somehow express the autocorrelation function of $|a(t)|^2$ as a function of $\phi_{aa}(\tau)$. $$ \begin{align} \phi_{|a|^2|a|^2}(\tau)&=\left(|a(t)|^2\star|a(t)|^2\right)(\tau)=\int_{-\infty}^{\infty}|a(t)|^2\cdot |a(t+\tau)|^2 \mathrm{d}t\\ &=\int_{-\infty}^{\infty} a(t)\cdot a^*(t)\cdot a(t+\tau)\cdot a^*(t+\tau) \mathrm{d}t \end{align} $$ So far I tried to somehow express the second integral using the first one, without success. I also tried to solve the problem in the fourier domain.
Is it in general possible to express the second autocorrelation in terms of the first one? Are there maybe any special assumptions to be made?
