Slowly varying function

Short description: Function in mathematics

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,

[math]\displaystyle{ \lim_{x \to \infty} \frac{L(ax)}{L(x)}=1. }[/math]

Definition 2. Let L : (0, +∞) → (0, +∞). Then L is a regularly varying function if and only if [math]\displaystyle{ \forall a \gt 0, g_L(a) = \lim_{x \to \infty} \frac{L(ax)}{L(x)} \in \mathbb{R}^{+} }[/math]. In particular, the limit must be finite.

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).

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

[math]\displaystyle{ f(x)=x^\beta L(x) }[/math]

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

[math]\displaystyle{ g(a)=a^\rho }[/math]

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 ≥ B the function can be written in the form

[math]\displaystyle{ L(x) = \exp \left( \eta(x) + \int_B^x \frac{\varepsilon(t)}{t} \,dt \right) }[/math]

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

[math]\displaystyle{ \lim_{x \to \infty} L(x) = b \in (0,\infty), }[/math]

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, nor 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

References

• Hazewinkel, Michiel, ed. (2001), "Karamata theory", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• 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, https://archive.org/details/regularvariation0000bing 
• 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 .