Let's say we have discrete-time stationary random signals with Gaussian PDF of mean value 0 and variance 1, whose individual signal values are uncorrelated.
For such a signal, how can we determine ACF and the PSD?
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ACF:
$$\begin{align} R_x[k] &= \lim_{N \to \infty} \tfrac{1}{2N+1} \sum\limits_{n=-N}^{N} x[n] \, x[n+k] \\ &= \mathbb{E}\Big\{ x[n] \, x[n+k] \Big\} \\ &= \sigma_x^2 \ \delta[k] \\ &= 1\ \delta[k] \\ \end{align}$$
PSD:
$$ S_x(e^{j\omega}) = \sum\limits_{n=-\infty}^{\infty} R_x[n] \, e^{-j \omega n} $$
You have described a discrete-time Gaussian white noise to a T.
For a white stationary Gaussian random process $n[k]$, the autocorrelation $R_n[k]$ is: $$ R_n[k] = \mathbb E \left( n[m]n[m+k] \right) = \sigma^2 \delta[k], $$ where $\sigma^2$ is the variance.
The power spectral density $P(\omega)$ is just the DTFT of the autocorrelation, which is: $$ P(\omega) = \sum_{k=-\infty}^\infty R_n[k] e^{j\omega k} = \sigma^2 $$ i.e., it is the same for all frequencies ("white noise").