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 and extreme value theory.

Basic definitions

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

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

Definition 2. Let $L : (0, + \infty) \to (0, + \infty)$ . Then $L$ is a regularly varying function if and only if $\forall a>0,g_{L}(a)=\lim _{x\to \infty }{\frac {L(ax)}{L(x)}}\in \mathbb {R} ^{+}$ . 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 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 $f : (0, + \infty) \to (0, + \infty)$ is of the form

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

where

$β$ is a real number,
$L$ 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 $x \geq B$ the function can be written in the form

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

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

If $L$ is a measurable function and has a limit

\lim _{x\to \infty }L(x)=b\in (0,\infty ),

then

L

is a slowly varying function.

For any $β \in R$ , the function $L (x) = log β x$ is slowly varying.
The function $L (x) = x$ is not slowly varying, nor is $L (x) = x β$ for any real $β \neq 0$ . However, these functions are regularly varying.

Notes

1 2 3 See (Galambos & Seneta 1973)
1 2 See (Bingham, Goldie & Teugels 1987).

References

Bingham, N.H. (2001) [1994], "Karamata theory", Encyclopedia of Mathematics, EMS Press
Bingham, N. H.; Goldie, C. M.; Teugels, J. L. (1987), Regular Variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge: Cambridge University Press, ISBN 0-521-30787-2, MR 0898871, Zbl 0617.26001
Galambos, J.; Seneta, E. (1973), "Regularly Varying Sequences", Proceedings of the American Mathematical Society, 41 (1): 110–116, doi:10.2307/2038824, ISSN 0002-9939, JSTOR 2038824.

[GaSe-1] 1 2 3 See (Galambos & Seneta 1973)

[BiGoTe-2] 1 2 See (Bingham, Goldie & Teugels 1987).