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 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 is called slowly varying if for all,
. Let. Then 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 is restricted to a compact interval.

Karamata's characterization theorem

. Every regularly varying function is of the form
where

is a real number,
is a slowly varying function.

Note. This implies that the function in has necessarily to be of the following form
where the real number is called the index of regular variation.

Karamata representation theorem

. A function is slowly varying if and only if there exists such that for all the function can be written in the form
where

is a bounded measurable function of a real variable converging to a finite number as goes to infinity
is a bounded measurable function of a real variable converging to zero as goes to infinity.

Examples

If is a measurable function and has a limit
For any, the function is slowly varying.
The function is not slowly varying, nor is for any real. However, these functions are regularly varying.