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If it is given that PSD of a random process is bandlimited to frequency $f_B$, then can we claim that any sample path of the random process is also bandlimited to $f_B$?

Intuitively, I always thought of PSD as spectrum of the random signal in some sense but mathematically, the relationship between PSD of the random process and Fourier transform of the random process is not clear to me. If the PSD is limited to $f_B$, then it is a statement on the autocorrelation function of the WSS process. Does this translate to every sample path of the random process being bandlimited to $f_B$ as well?

Edit: For clarity, I'm considering a discrete time random process $X[n]$.

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  • $\begingroup$ I've answered the question making a strong assumption on what your "sample path" is. If you could mathematically define what a sample path is in your case, this might help! $\endgroup$
    – Marcus Müller
    Aug 4, 2022 at 10:28
  • $\begingroup$ The answer is no. Like a realization of the standard normal random variable can be different from zero. $\endgroup$
    – AlexTP
    Aug 4, 2022 at 20:13
  • $\begingroup$ Autocorrelations and power spectral densities are defined in terms of mean-square convergence which is not necessarily the same as almost sure convergence which is what you are asking about. So the best answer is "We can't be sure that the sample path is almost surely band-limited". $\endgroup$
    – Dilip Sarwate
    Aug 14, 2022 at 15:43

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Just real quick: Relationship of PSD and Fourier Transform: Only exists if the process is WSS, as per the Wiener-Khinchin Theorem, and then the PSD is the magnitude square of the Fourier transform in expectation.

Since WSS says that for any two times the the autocorrelation only depends on the difference between these times, this especially holds for a subset of times, i.e., your sample path.

I'll have to inquire more about what your "sample path" is, but if it's taking samples at equally spaced times, then yes, this just "stretches" the Fourier transform, so your $f_B$ bandlimiting (for the time-continuous or $f_\text{sample}$-sampled time-discrete original signal) becomes scaled by the inverse of these sampling intervals.

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  • $\begingroup$ Hi, by sample path, I meant a realization of the random process. I have updated the question as well. I think your 2nd and 3rd paragraph are a little bit different from the question. Can you please elaborate on the first paragraph? It seems to answer my question - if the expected energy is 0 outside $f_B$, then all realizations of the random process are bandlimited by $f_B$ almost surely? $\endgroup$
    – Black Jack 21
    Aug 4, 2022 at 16:50
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    $\begingroup$ hm, but that "sample path" is then just a realization of your stochastic process, i.e. the signal. As such, it's a realization, which, strictly speaking, doesn't have a PSD (that's a property of the process as a whole); in expectation, yes, if the process is WSS, the expectation of the Fourier transform of realizations is identical to the PSD. $\endgroup$
    – Marcus Müller
    Aug 4, 2022 at 16:58

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