# Auto-correlation of absolute squared stochastic process

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?

• your first formula is missing an Expectation operator! – Marcus Müller Dec 18 '20 at 17:07
• Your definition of the autocorrelation function of a stochastic process is wrong for more reasons than the one mentioned by Marcus Muller. – Dilip Sarwate Dec 18 '20 at 19:14
• avoiding getting into a discussion here in the comments: a) from the existence of an autocorrelation function of $a$ you can't infer the existence of one of $\lvert a\rvert^2$; not every integrable function is absolute integrateable. b) You can't base much sensible consideration on a false definition; you're applying the definition of autocorellation of deterministic signals to stochastic ones; that won't work; your definition is really $\phi_{aa}(\tau) = \mathbb E\left[ a(t) a^*(t+\tau)\right]$; that expectation operator is not implementable as an integral across all times in general! – Marcus Müller Dec 19 '20 at 11:09