I am trying to better understand the spectral representation of stochastic processes. From the book "Spectral Analysis for physical applications" by Walden and Persival:
The spectral representation theorem for discrete parameter stationary processes states: "Let $\{X_t\}$ be a [...] discrete parameter stationary process with zero mean. There exists an orthogonal process, $\{Z(f)\}$, defined on the interval [—1/2,1/2], such that
$$ X_t = \int_{-1/2}^{1/2}e^{i2\pi ft}dZ(f) $$ "
I was wondering how would $Z(f)$ look for $X_t$ a white process (i.e., the power spectrum density is flat). I came up with the following possibility, and was wondering if this is correct and/or if there are some better/other possibilities as to how $Z$ could look:
$$ Z(-1/2) := 0\\ dZ(f_j) := 1/\sqrt{N} e^{i2 \pi \phi_j} \quad \text{for} \; f_j = j/(2N),\, j=-N,...,N $$
with $\phi_j$ uniform random variables on $[-\pi, \pi]$. Then taking the limit for $N\rightarrow \infty$ would give a white process?