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.

Basic definitions

. A measurable function L : (0, +∞) → (0, +∞) is called slowly varying (at infinity) if for all a > 0 , . Let L : (0, +∞) → (0, +∞) . Then L is a regularly varying function if and only if. In particular, the limit must be finite. These definitions are due to Jovan Karamata.

Basic properties

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.

Uniformity of the limiting behaviour

. The limit in and is uniform if a is restricted to a compact interval.

Karamata's characterization theorem

. Every regularly varying function f : (0, +∞) → (0, +∞) is of the form where Note. This implies that the function g(a) in has necessarily to be of the following form where the real number ρ is called the index of regular variation.

Karamata 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) 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.

Examples

β ∈ R , the function is slowly varying. is not slowly varying, nor is for any real β ≠ 0 . However, these functions are regularly varying.