So to add more details, the an undefined PSD for the noise generated means that I could have any shape of PSD and just hope that my PLL filters out the noise from the VCO noise model. The below given examples are different possibilities of having the same sigma but different PSDs. In case anyone would like to comment on anything I'm missing, please do as it is not of any use for a designer to spend time using an unknown or unrealistic AMS or verilogA model in his PLL  design.

Yes. I agree it is used in time domain model. But if i analyze the noise profile of the model in frequency domain by taking FFT of the samples, what kind of spectrum do you think I'll get? In fact the sigma is the input of the noise model while we don't know what kind of PSD it has.

Since I received no reply from anyone regarding this, I did my own little research to find topics like maximum length sequence. I believe there are algorithms which are used to generate the sequences. The autocorrelation of the sequence is defined. In other words the power spectral density is closer to white based on the length. I think it's a topic of study by itself. There is also this book called shift register sequences which may be of interest to anyone who stumbles across this post and same doubt.

Hi Ken,
I completely agree to the point that without time there is no point talking about the PSD of a sequence. My first reference to some understanding as to why the random numbers sequence when used as a waveform should work is from the below links :

https://en.wikipedia.org/wiki/Maximum_length_sequence
https://www.amazon.com/Shift-Register-Sequences-Solomon-Golomb/dp/0894120484

Nope. I could literally have any shape of PSD which gives me the same area under the curve (variance) for the gaussian curve you're talking about. My question was as to how do we know if the PSD or the power of the resultant signal generated using those sequences have white PSD. Rather than PSD, such sequences are characterized by Autocorrelation function.

Hi Ken,
Actually I only have a rough mathematical intuition behind how random number generated maybe roughly white till f<<1/Ts. To get to there, I tried picturing how I could get finite valued samples from white noise theoretically. Sampling white noise using a dirac delta will have infinite PSD. But then an real oscillator couldn't do a delta sampling. I then felt maybe the Impulse Sensitivity Funciton(ISF) itself can be assumed to be the mixing quantity than dirac delta. This does low pass filtering preventing infinite-magnitude spectrum even after aliasing, given the sum of square of fourier series component of periodic ISF is finite. The cut off of the filter is inversely proportional to how thin the ISF pulse is. The pulse width is anyway < Ts.But still we have continuous waveforms rather than discrete kronecker delta which maybe roughly equivalent when continuous time simulations are being performed. This averaging over a period decides the whiteness for f<<1/Ts. I would be delighted to know if this is a correct direction of thinking.

DIRAC DELTA
I currently find it hard to picture sampling of white noise by dirac delta train as :
1. It results in infinite PSD or Infinite Variance (All hell breaks lose)
2. Dirac delta trains rather than finite valued-finite width samples.

PULSE SAMPLING
ISF for my perspective looked like a gate which periodically opens and closes allowing noise to get sampled for an oscillator like a pulse. Pulse sampling was chosen to avoid infinite aliasing because if we had a periodc ISF train, then its fourier coefficients die down at high freq. Sum of Squares of the coeffiecients will be the multiplication factor on the white noise in the band. But then any pulse couldn't be it, but ISF.

BAND LIMITING
Definitely pulse sampling followed by the inverter's RC LPF-hold(say for a ring oscillator) results in a band determined by the RC (< Ts). Ignoring the above two para, we have a finite valued sample valid for one time period Ts and band limited to f<1/Ts.

In case you think, this thought could be erronous, let's not pursue it. I value your precious time. I'm also trying to construct a theory.