The Fréchet distribution: $$\Phi_\alpha(x)=\begin{cases}0 & & x\leq 0,\\[6pt]e^{-x^{-\alpha}} & & x>0,\end{cases}$$

is regularly varying as stated here (page 19):

It is not difficult to see that the Fréchet distribution is regularly varying with index $\alpha.$

Now from the same source (page 7), a function is regularly varying (at infinity) if

$$\lim_{x\to \infty}\frac{f(tx)}{f(x)}=t^\alpha,$$

and if $\alpha=0$ it is called slowly varying (at infinity). Since $t^0=1,$ this coincides with the definition in Wikipedia of slowly varying functions: a slowly varying function in which the relative differences in the tail is equal to zero: $$\lim_{x\to \infty} \frac{f(tx) - f(x)}{f(x)}=0.$$

Is the author saying that for the Fréchet distribution:

$$\lim_{x\to\infty}\frac{e^{-(\alpha x)^{-\alpha}}}{e^{-x^{-\alpha}}}=\alpha$$

This doesn't seem correct. In what sense is it correct, and what does it imply in terms of understanding the behavior of the distribution?

  • $\begingroup$ Your second limit is not an application of the first, so possibly you have made a typographical error. $\endgroup$
    – whuber
    Jun 6 at 22:14

Definition 1.3.1 (p. 11) says that $\bar{\Phi}_\alpha$ (right distribution tail) needs to be regularly varying with $-\alpha$. That is true (change of variable and L'Hôpital, $\alpha >0$):

$$ \lim_{x\rightarrow \infty} \frac{1- e^{-(tx)^{-\alpha}}}{1-e^{-x^{-\alpha}}} = \lim_{y\rightarrow 0} \frac{1- e^{-t^{-\alpha}y}}{1-e^{-y}} = \lim_{y\rightarrow 0} \frac{t^{-\alpha}e^{-t^{-\alpha}y}}{e^{-y}} = t^{-\alpha}.$$

  • $\begingroup$ It's therefore the survival function (right tail) that is regularly varying, not the actual distribution? $\endgroup$ Jun 6 at 22:28
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    $\begingroup$ No. Second limit gives $\frac{1}{1}=1$ by plugging in $y=0$. It's not a L'Hospital case. like $\frac{0}{0}$. $\endgroup$
    – ir7
    Jun 6 at 22:49
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    $\begingroup$ @Ben Sorry, but French is only the 15th most spoken language, pathetic compared to English ranked third :). wiki: "In mathematics, more specifically calculus, L'Hôpital's rule or L'Hospital's rule (French: [lopital], English: /ˌloʊpiːˈtɑːl/, loh-pee-TAHL) provides a technique to evaluate limits of indeterminate forms." $\endgroup$
    – ir7
    Jun 6 at 23:31
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    $\begingroup$ Fair enough; feel free to change it back! ; ) $\endgroup$
    – Ben
    Jun 6 at 23:32
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    $\begingroup$ I have just done some more research and it looks like the spelling issue is more complicated than I thought. It appears that the spelling actually used to be l'Hospital (with the 's') and then there was a reform to the French language in the 18th century to drop the silent 's' in words like this. So in the Marquis's own time his name probably would have been spelled with the 's', but no longer. In a historical edition in 1768 his name was spelled without the 's'. How very interesting! $\endgroup$
    – Ben
    Jun 6 at 23:41

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