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In an earlier question I asked Why are processes with stationary independent increments nonstationary?, i.e. why are Lévy processes nonstationary, and saz gave a nice proof using characteristic functions that fully answers the question.

Looking over the proof, though, I realized that there's something more that I would like, beyond a conclusive reason for believing that Lévy processes are nonstationary, which saz has provided.

Is there a way to see, intuitively, why stationary increments must turn into nonstationarity? How is nonstationarity built from stationary increments? What is the relationship between them?

Maybe there are better ways to ask this question, but now I'm asking "why" in the sense of wanting an intuitive understanding, rather than asking for a bare reason to believe that it is so. [Is there a better way to say this? I know that this isn't the first question of this kind in Math.SE.]

If there is a good answer, it needn't be rigorous, as saz has already provided a rigorous answer.

(If this is too close to the earlier question linked above, and I should not have posted this one, let me know and I will delete this one.)

Mars
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  • In case the process has finite variance : notice that variance add up over time. Otherwise compute the Fourier transform/characteristic function (but this is no more intuitive perhaps, and is anyway the content of the previous post). – Olivier Aug 08 '19 at 14:44
  • Hmm will think about both ideas, @Olivier. The proof in the answer did use Fourier transforms, and I have gotten insight from thinking about their meaning in the past, but this proof used them in a way that, for me, didn't provide insight. Some characteristic function tricks are like that. Maybe I am just not yet thinking about the proof from the right perspective, though. So your suggestion about Fourier transforms is not irrelevant. – Mars Aug 08 '19 at 14:51

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