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,

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

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.^[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 finite interval.

Karamata's characterization theorem

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

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

where

• β ≥ 0 is a non negative 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 non negative 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 ≥ 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 has a limit

$\lim_{x \to \infty} L(x) = b \in (0,\infty),$

then L is a slowly varying function.

• For any β ∈ R, the function L(x) = log^β x is slowly varying.
• The function L(x) = x is not slowly varying, neither is L(x) = x^β for any real β ≠ 0. However, these functions are regularly varying.

See also

• Analytic number theory
• Hardy–Littlewood tauberian theorem and its treatment by Karamata

Notes

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

References

• Bingham, N.H. (2001), "Slowly varying function", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Bingham, N. H.; Goldie, C. M.; Teugels, J. L. (1987), Regular Variation, Encyclopedia of Mathematics and its Applications 27, Cambridge: Cambridge University Press, ISBN 0-521-30787-2, MR MR0898871, 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.