# Relation between frequency spectrum and PDF of a random variable

I have a random variable that is being generated according to some probability distribution function (e.g. a Gaussian PDF).

• When looking at the frequency spectrum of the generated data does the knowledge of the exact distribution function have any constraints/give any information about the frequency spectrum of the samples?

• In other words, would the PDF of a random variable constrain the possible frequency spectra the resultant random process may take?

• Is this a reasonable question?

• Given that random processes are random(aperiodic) and independent (uncorrelated and thus unfiltered), I would say the Fourier transform always converges to a constant. It's up to someone else to prove this and to inspect whether I made a fundamental mistake. Good question. – Dole Jul 3 '16 at 10:08
• @Dole: It's a big misunderstanding that random processes are "uncorrelated". Of course there can (and there usually is) correlation between different samples of a random process. Furthermore, when you say "the Fourier transform always converges to a constant", what do you mean? The Fourier transform of what? – Matt L. Jul 3 '16 at 10:18
• @MattL. I don't think it's misunderstanding that random processes as described in the question are uncorrelated. Random processes generated by independent random variables are independent and thus uncorrelated. As for the fourier transform I mean the magnitude spectra (not including DC) of the realization of the random process, for example uniform distribution noise process, gaussian noise process... etc. – Dole Jul 3 '16 at 11:45
• @Dole: But the OP didn't say that the RVs are independent, that's why I pointed out that uncorrelatedness does not follow from the facts given in the question. Concerning the Fourier transform, usually you cannot assume that the Fourier transform of a realization of a random process even exists. That's why we use the power spectral density (in the case of stationary processes), which is the Fourier transform of the auto-correlation function. – Matt L. Jul 3 '16 at 12:24
• I think this is a very reasonable question, but you should re-formulate the last sentence: "the possible frequency spectra the random variable may take" does not make sense, because a random variable doesn't have a spectrum. What you probably mean is the random process that you generate as a sequence of random variables. – Matt L. Jul 3 '16 at 12:27

Let's take this from the bottom:

Is this a reasonable question?

Of course!

In other words, would the PDF of a random variable constrain the possible frequency spectra the resultant random process may take?

That depends. For example, if the PDF of your RV was a Dirac impulse, i.e. your RV was actually a constant, then, obviously, yes, that will influence spectral aspects of your random process.

Now, in general, a random process is random, i.e. one can not say what its spectrum is going to be (or whether that is possible to define at all).

Hence, you can only do things like considering a specific statistical/stochastic property the random process has.

That, for example, and also most likely closest to what you're looking for, would be the Power Spectral Density (PSD).

Now, intuitively, one would just say that you could take infinitely many realizations of a random process, transform them to the frequency domain (using the Fourier transform), and calculate the expectation value of these transform's magnitude squares.

However, as Matt hinted at, it's really not easy to show that this transform, or the expectation value, are well-defined for some types of processes (or if you can't make some strong assumptions on the process' properties).

Hence, mathematically, you take the elegant detour of first calculating the autocorrelation, which, again, is a property of the process, and Fourier transforming that.

For particularly well-behaved processes (mainly: Your process needs to be wide-sense stationary), this identical to the aforementioned expectation of the magnitude squared Fourier transform across all realizations; that's the Wiener-Chintschin theorem, which I mentioned in another answer that might be of relevance to you.

The point here is: Together with some statements on the autocorrelation of the process, and by observing the PDFs at fixed times, you might come to some statements on the process' PSD.

When looking at the frequency spectrum of the generated data does the knowledge of the exact distribution function have any constraints/give any information about the frequency spectrum of the samples?

No. This is exactly the case where lack of ergodicity means that you can't derive much from a time ensemble.

• I am pretty much a beginner to stochastic processes, so forgive my ignorance but I do not understand how you came to the latter half of the premise Together with some statements on the autocorrelation of the process, and by observing the PDFs at fixed times, you might come to some statements on the process' PSD. Would appreciate it if you could elaborate a little. Thanks for your response! – Television Jul 4 '16 at 4:19
• I think you should familiarize yourself with the term of ergodic processes, which is a very special case of processes, but that will probably give you the insight of all the cases where a time ensemble is not representative for an ensemble in state space (i.e. over different realizations) – Marcus Müller Jul 4 '16 at 8:02
• Yes, on first look it seems very fundamental to practical stochastic processes, so I will have to familiarize. Thx for the discussion, have a nice day! – Television Jul 5 '16 at 4:24