View source for De Haan theory

A branch of the theory of regular variation, initiated in 1930 by the Yugoslav mathematician J. Karamata. This theory studies asymptotic relations of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d1100301.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

and has many interesting properties. For instance, under mild conditions (e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d1100302.png" /> measurable) one has: i) uniform convergence on compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d1100303.png" />-sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d1100304.png" />; ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d1100305.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d1100306.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d1100307.png" /> is said to vary regularly with index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d1100308.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d1100309.png" /></td> </tr></table>

One can work instead with the more general relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d11003010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d11003011.png" /> of which (the  "Karamata case" ) reduces to (a1) on exponentiation and change of notation. This theory goes back to R. Bojanic and Karamata in technical reports of 1963, but was studied extensively by L. de Haan [[#References|[a2]]] in his 1970 thesis and subsequently. One finds that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d11003012.png" /> varies regularly: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d11003013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d11003014.png" />, nothing new is obtained, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d11003015.png" /> leads to a new function class, the de Haan class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d11003016.png" />, a proper subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d11003017.png" />. It, and the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d11003018.png" /> containing inverses of its members, have been studied in detail (uniform convergence theorem as above, etc.); see e.g. [[#References|[a1]]], Chap. 3.

The resulting theory (de Haan theory) is both a direct generalization of the [[Karamata theory|Karamata theory]] above and what is needed to fill certain gaps, or boundary cases, in Karamata's main theorem. The original motivation was probabilistic, arising (following the disastrous floods in the Netherlands in 1953) in the study of extreme values (of sea levels, wind speeds, etc.). This study of extremes goes back to R.A. Fisher and L.H.C. Tippett in 1928, and was known to lead to three types of limit distribution, the [[Weibull distribution|Weibull distribution]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d11003019.png" />, (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d11003020.png" />), the Fréchet distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d11003021.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d11003022.png" />) and the Gumbel distribution (or double-exponential distribution) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d110/d110030/d11003023.png" />. It was known from earlier work, using [[Karamata theory|Karamata theory]], which distributions lead to Weibull or Fréchet limits. The original motivation for de Haan theory, and its first achievement, was a correspondingly complete solution for the case of Gumbel limits ([[#References|[a2]]]; [[#References|[a1]]], §8.13). This case is particularly important in applications, as it includes so many commonly occurring cases, including the normal distributions (cf. [[Normal distribution|Normal distribution]]). It is also needed for related problems such as those of the statistics of records. An extensive theory of the probability and statistics involved in extremes has now been developed (see e.g. [[#References|[a3]]], [[#References|[a4]]]). This is largely complete in the classical case (independent readings, one dimension), and has been much studied also in more general settings (dependence, higher dimensions).

De Haan theory is also powerful within pure mathematics. For instance, it leads to a particularly rapid proof of the prime number theorem (see e.g. [[#References|[a1]]], §6.2; see also [[De la Vallée-Poussin theorem|de la Vallée-Poussin theorem]]), among many other things.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.H. Bingham,   C.M. Goldie,   J.L Teugels,   "Regular variation" , ''Encycl. Math. Appl.'' , '''27''' , Cambridge Univ. Press  (1989)  (Edition: Second)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. de Haan,   "On regular variation and its application to the weak convergence of sample extremes" , ''Tracts'' , '''32''' , Math. Centre , Amsterdam  (1970)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.R. Leadbetter,   G. Lindgren,   H. Rootzén,   "Extremes and related properties of random sequences and processes" , Springer  (1983)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S.I. Resnick,   "Extreme values, regular variation and point processes" , Springer  (1987)</TD></TR></table>