I have the following problem:
Let $\theta$ be a i.i.d. random variable with distribution $U[-\pi,\pi]$, $Wt$ the standart Wiener process and $\sigma$ a real number. Define the stochastic process $$Y_t = \cos(\omega t + \sigma Wt +\theta)$$
a bounded noise with mean
$$\mu=\mathbb{E}[Yt]=0$$
and correlation function
$$R=\mathbb{E}[Y_t,Y_{t+\tau}]=\frac{1}{2}\cos(\omega \tau)e^{\frac{(-\sigma^2)}{2}|\tau|}$$
While I was obtaining the Fourier expansion for this process,
$$Y_t\approx\hat{Y_t}=\sum_{n=0}^\infty c_n e^{-in\omega t}$$
I got stuck trying to understand the random variable $c_n$, which in various textbooks is simply defined "from the integral":
$$c_n=\int_{-T}^T Y_t e^{-in\omega t}dt$$
What does this mean? What information this integral possess? How can I obtain a random variable from an integral of a process that I am trying to discretize in first place? How can I obtain its density or any other distribution from this expression?
Bonus question: is there an analytic Karhunen-Loeve expansion for this particular proccess? I have tryied to get through the Fredholm integral, but without success.
Any help is deeply appreciated.
Edit:
$\omega$ is the frequency of the process;
$\sigma$ is the standart deviation;
$\theta$ is sampled only once at $t_0$, making it a initial phase angle. It is necessary to be included so $R$ stay bounded and the process can be discretized by a Fourier type series.