This is rather a straightforward question.
Suppose a stochastic process is wide sense stationary (WSS) & it is ergodic as well. Is the power spectral density (PSD) say of a realisation, $\hat{S}(\omega)$, of this process equivalent to PSD of the process $S(\omega)$ itself? That is, $\hat{S}(\omega) \equiv S(\omega)$
In this sense, isn't PSD then deterministic? If so, I'm unable to interpret, the following statements
assuming that the random process x(t) is a WSS ergodic normal random process with zero mean and variance $\sigma^2$
$x(t) = \sum_{n=0}^{\infty} a_n cos(\omega_n t + \epsilon_n)$ where a_n is a random variable with a normal distribution with zero mean & some variance.
Then at another point it is asserted that $\frac{a_i^2}{2} = S(\omega_i)\Delta \omega_i$.
If $S(\omega)$ is deterministic then how is a random variable asserted to be equal to a deterministic one.