I have two integrals
$$\int_{0}^{\infty} \mathbb{E}[\cos 2 \psi(0) \cos 2 \psi(z)] d z$$
$$\int_{0}^{\infty} \mathbb{E}[\sin 2 \psi(0) \sin 2 \psi(z)] d z$$
where $\psi(z)$ is randomly varying and can be modelled by a stationary and ergodic process whose stationary distribution is uniform over $[0,2\pi].$ One can then assume that $\psi(z)$ is a Markov process on the torus with generator satisfying the Fredholm alternative.
My first question is, can one equate these two integrals?
My second question is, are the two following facts true?
$$\int_{0}^{\infty} \mathbb{E}[\cos 2 \psi(0) \sin 2 \psi(z)] d z = 0$$
$$\int_{0}^{\infty} \mathbb{E}[\sin 2 \psi(0) \cos 2 \psi(z)] d z = 0$$
Orthogonal functions pop into my mind but this is different since $\psi$ is a Markov process.
Since $\psi(z)$ has an invariant distrubution over $[0,2\pi]$, it can be distributed as $\psi(0) + \theta, \psi(z)+\theta$ where $\theta$ is a fixed angle, the integrals equate.