Slowly varying function
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, and have found several important applications, for example in probability theory.
Definition 1. A 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.
Regularly varying functions have some important properties: 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 behaviour
Karamata's characterization theorem
Theorem 2. Every regularly varying function f is of the form
- 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 theorem
Theorem 3. A function L is slowly varying if and only if there exists B > 0 such that for all x ≥ B the function can be written in the form
- η(x) is a bounded measurable function of a real variable converging to a finite number as x goes to infinity
- ε(x) is a bounded measurable function of a real variable converging to zero as x goes to infinity.
- If L has a limit
- then L is a slowly varying function.
- For any β ∈ R, 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.
- Bingham, N.H. (2001), "Slowly varying function", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, 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 MR0898871, 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.