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I was wondering why I always read that the decay of the ACF of an AR(p) process should be exponential. I am confused because the theoretical ACF of a stationary AR(1) process $y_{t} = \phi y_{t-1} + \epsilon_{t}$, $\mid\phi\mid < 1$ is $\phi^{h}$ where $h$ is the lag. So this is a power law decay. Why do I read/see so many times the decay is exponential?

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3 Answers 3

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$f(h) =\phi^h$ is an exponential function.

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The magnitude of the ACF is an exponential function in $h$:

$$|\phi|^h = \exp( \log (|\phi|^h)) = \exp( h \log |\phi|).$$

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  • $\begingroup$ Your manipulation is correct but it give the erroneous impression that the operator “exp” is needed. We can speak about exponential function without the Nepero number. Indeed, for example, $f(x)=5^x$ is a proper exponential function. $\endgroup$
    – markowitz
    Apr 10, 2021 at 14:56
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    $\begingroup$ The usual definition of $5^x$ over the real numbers is $5^x \equiv \exp(x \log 5)$ (see e.g., discussion here). If one takes that as the definition, then $\exp$ certainly is needed. While there are other ways to define exponents of irrational numbers (e.g., through limits of rational exponents), this is the standard definition and seems to me to be the least cumbersome. Can you explain what you understand $f(\sqrt{2})=5^{\sqrt{2}}$ to mean, if not via use of the $\exp$ function? $\endgroup$
    – Ben
    Apr 10, 2021 at 21:15
  • $\begingroup$ My comment come from the fact that non only $e^x$ but also $a^x$ is an exponential function in $x$ for any constant $a>0$. Some guys erroneously hesitate to consider also the second as exponential function. The name “exp” can lead in confusion here; maybe the asker fallen in confusion here too. Said that, you said that $a^x=exp(x*ln(a))$ is a definition. I’m not sure. When I was student I used serious math book that do not mentioned that and also in many places is currently not used. $\endgroup$
    – markowitz
    Apr 11, 2021 at 8:49
  • $\begingroup$ The argument that you invoke seems be interesting for arithmetic lovers only. As you said, and as the discussion linked shown, “limits of rational exponents” can be used there. In my view this tool is the relevant there. In practice we use some approximation in situation like that; after all even for compute $\sqrt(2)$ we use some approx. Your formula can help for computation, sometimes, but is not a definition and … finally is not needed. Now, you ask, what I mean for $f(\sqrt(2))=5^{\sqrt(2)}$, this is simply a real number for me, we can end there. $\endgroup$
    – markowitz
    Apr 11, 2021 at 8:52
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No, thinking in $h$, this is not power-law. It would be if it was something like $h^\phi$. Therefore, autocorrelation, $\phi^h$, is referred as exponential.

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