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

$$ \tag{a1 } { \frac{f ( \lambda x ) }{f ( x ) } } \rightarrow g ( \lambda ) \in ( 0, \infty ) ( x \rightarrow \infty ) , \forall \lambda > 0, $$

and has many interesting properties. For instance, under mild conditions (e.g., $ f $ measurable) one has: i) uniform convergence on compact $ \lambda $- sets in $ ( 0, \infty ) $; ii) $ g ( \lambda ) \equiv \lambda ^ \rho $ for some $ \rho $. In this case $ f $ is said to vary regularly with index $ \rho $:

$$ f \in R _ \rho . $$

One can work instead with the more general relation

$$ \tag{a2 } { \frac{f ( \lambda x ) - f ( x ) }{g ( x ) } } \rightarrow h ( \lambda ) \in \mathbf R ( x \rightarrow \infty ) , \forall \lambda > 0, $$

the case $ g \equiv 1 $ 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 [a2] in his 1970 thesis and subsequently. One finds that $ g $ varies regularly: $ g \in R _ \rho $. If $ \rho \neq 0 $, nothing new is obtained, but $ \rho = 0 $ leads to a new function class, the de Haan class $ \Pi $, a proper subset of $ R _ {0} $. It, and the class $ \Gamma $ containing inverses of its members, have been studied in detail (uniform convergence theorem as above, etc.); see e.g. [a1], Chap. 3.

The resulting theory (de Haan theory) is both a direct generalization of the 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 $ \Psi _ \alpha $, ( $ \alpha > 0 $), the Fréchet distribution $ \Phi _ \alpha $( $ \alpha > 0 $) and the Gumbel distribution (or double-exponential distribution) $ \Lambda ( x ) = { \mathop{\rm exp} } ( - e ^ {- x } ) $. It was known from earlier work, using 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 ([a2]; [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). 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. [a3], [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. [a1], §6.2; see also de la Vallée-Poussin theorem), among many other things.

References

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