# 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,[1][2] and have found several important applications, for example in probability theory.

## Basic definitions

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

${\displaystyle \lim _{x\to \infty }{\frac {L(ax)}{L(x)}}=1.}$

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

${\displaystyle g(a)=\lim _{x\to \infty }{\frac {L(ax)}{L(x)}}}$

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

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

## Basic properties

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 behaviour

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

### Karamata's characterization theorem

Theorem 2. Every regularly varying function f is of the form

${\displaystyle f(x)=x^{\beta }L(x)}$

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

${\displaystyle g(a)=a^{\rho }}$

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 xB the function can be written in the form

${\displaystyle L(x)=\exp \left(\eta (x)+\int _{B}^{x}{\frac {\varepsilon (t)}{t}}\,dt\right)}$

where

## Examples

• If L has a limit
${\displaystyle \lim _{x\to \infty }L(x)=b\in (0,\infty ),}$
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.