My question is this:
Do the 'amplitudes' of a PSD depend on the number of samples on which it is based?
One definition of the PSD I have found is: $$ P_{xx}(\Omega_m) = \frac{X(\Omega_m)\cdot\bar{X}(\Omega_m)}{N}=\frac{|X(\Omega_m)|^2}{N} $$ With $N$ being the number of samples and $X(\Omega_m)$ being the $m^{ \rm th}$ point in the DFT of the time series. Based on this definition, the PSD values will increase with increasing $N$.
However, in other places (e.g. here ) I am being told that the strength of the PSD is that it does not vary with varying frequency resolution.
I have found that the definition above obeys the Wiener-Khintchine relation, when the autocorrelation is: $$ \hat{R}_{xx}(k) = \frac{1}{N}\cdot\sum_{n=0}^{N-1}{x[n]\cdot x[n+k]} $$ So far I think I have found five different definitions of the PSD. What is "the right one" in your opinion and does it depend on sample size?