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.

Basic definitions

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

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

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

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

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

These definitions are due to Jovan Karamata.

Note. In the regularly varying case, the sum of two slowly varying functions is again slowly varying function.

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 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 : (0,+∞) → (0,+∞) is of the form

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

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

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

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

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

where

Examples

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