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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The magnitude of the ACF is an exponential function in $h$:
$$|\phi|^h = \exp( \log (|\phi|^h)) = \exp( h \log |\phi|).$$
Ben
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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. – markowitz Apr 10 '21 at 14:56
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The usual *definition* of $5^x$ over the real numbers is $5^x \equiv \exp(x \log 5)$ (see e.g., discussion [here](https://math.stackexchange.com/questions/55068/)). 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? – Ben Apr 10 '21 at 21:15
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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. – markowitz Apr 11 '21 at 08:49
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for example see: https://www.mathsisfun.com/sets/function-exponential.html; http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U18_L1_T1_text_final.html; https://tutorial.math.lamar.edu/classes/alg/expfunctions.aspx; https://en.wikipedia.org/wiki/Exponential_function – markowitz Apr 11 '21 at 08:51
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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. – markowitz Apr 11 '21 at 08:52
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If we want an explicit number we achieve, by approx, something like $9,7385…$. Moreover note that your definition is problematic also because it is circular, define the exponential function with another exponential function. Moreover it do not solve all computational problems, indeed in situation like $f(x)=\sqrt(2)^x$ in $x=\sqrt(2)$ we have $\sqrt(2)^{\sqrt(2)}$ your formula become $exp(\sqrt(2)*ln(\sqrt(2)))$ … approximation remain needed. – markowitz Apr 11 '21 at 08:56
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With respect, I don't find your views here to be compelling. The definition I have given is standard and non-circular. The standard (and arguably simplest) approach is to define the exponential function $\exp$ and logarthmic function $\log$ by power series, and then define the broader class of expontial functions as I have stated. Moreover, since the OP is specifically confused about this very issue, showing the explicit expansion in terms of the $\exp$ function makes the translation clear. – Ben Apr 11 '21 at 12:42
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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.
gunes
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