Slowly varying function
In real analysis, a branch of mathematics, a slowly varying function is a function resembling a function converging at infinity. A regularly varying function resembles a power law function near infinity. Slowly varying and regularly varying functions are important in probability theory.
A function L: (0,∞) → (0, ∞) is called slowly varying (at infinity) if for all a > 0,
If the limit
is finite but nonzero for every a > 0, the function L is called a regularly varying function.
- 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, they are regularly varying.
Some important properties are (Galambos & Seneta 1973):
- The limit in the definition is uniform if a is restricted to a finite interval.
- Karamata's characterization theorem: every regularly varying function is of the form x βL(x) where β ≥ 0 and L is a slowly varying function. That is, the function g(a) in the definition has to be of the form aρ; the number ρ is called the index of regular variation.
- Representation theorem: 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
- where η(x) converges to a finite number and ε(x) converges to zero as x goes to infinity, and both functions are measurable and bounded.
- Bingham, N.H. (2001), "k/k110030", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Bingham, N.H.; Goldie, C.M.; Teugels, J.L. (1989), Regular Variation, Cambridge University Press.