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In real analysis, a branch of mathematics, a slowly varying function is a function of a real variable whose behaviour at infinity is in some sense similar to the behaviour of a function converging at infinity. Similarly, a regularly varying function is a function of a real variable whose behaviour at infinity is similar to the behaviour of a power law function (like a polynomial) near infinity. These classes of functions were both introduced by Jovan Karamata,[1][2] and have found several important applications, for example in probability theory.

Basic definitionsEdit

Definition 1. A measurable function L : (0,+∞) → (0,+∞) is called slowly varying (at infinity) if for all a > 0,

 

Definition 2. A function L : (0,+∞) → (0,+∞) for which the limit

 

is finite but nonzero for every a > 0, is called a regularly varying function.

These definitions are due to Jovan Karamata.[1][2]

Note. In the regularly varying case, the sum of two slowly varying functions is again slowly varying function.

Basic propertiesEdit

Regularly varying functions have some important properties:[1] a partial list of them is reported below. More extensive analyses of the properties characterizing regular variation are presented in the monograph by Bingham, Goldie & Teugels (1987).

Uniformity of the limiting behaviourEdit

Theorem 1. The limit in definitions 1 and 2 is uniform if a is restricted to a compact interval.

Karamata's characterization theoremEdit

Theorem 2. Every regularly varying function f : (0,+∞) → (0,+∞) is of the form

 

where

  • β is a real number, i.e. β ∈ R
  • L is a slowly varying function.

Note. This implies that the function g(a) in definition 2 has necessarily to be of the following form

 

where the real number ρ is called the index of regular variation.

Karamata representation theoremEdit

Theorem 3. A function L is slowly varying if and only if there exists B > 0 such that for all xB the function can be written in the form

 

where

ExamplesEdit

  • If L has a limit
 
then L is a slowly varying function.
  • For any βR, the function L(x) = logβ x is slowly varying.
  • The function L(x) = x is not slowly varying, neither is L(x) = xβ for any real β ≠ 0. However, these functions are regularly varying.

See alsoEdit

NotesEdit

ReferencesEdit

  • Bingham, N.H. (2001) [1994], "Slowly varying function", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  • Bingham, N. H.; Goldie, C. M.; Teugels, J. L. (1987), Regular Variation, Encyclopedia of Mathematics and its Applications, 27, Cambridge: Cambridge University Press, ISBN 0-521-30787-2, MR 0898871, Zbl 0617.26001
  • Galambos, J.; Seneta, E. (1973), "Regularly Varying Sequences", Proceedings of the American Mathematical Society, 41 (1): 110–116, doi:10.2307/2038824, ISSN 0002-9939, JSTOR 2038824.