Slowly varying function

Basic definitions

. 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.

. Let L : (0, +∞) → (0, +∞). 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 .

Uniformity of the limiting behaviour

. The limit in and is uniform if a is restricted to a compact interval.

Karamata's characterization theorem

. Every regularly varying function f : (0, +∞) → (0, +∞) 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 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

. 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 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 ∈ R, the function is slowly varying.
• The function is not slowly varying, nor is for any real *β *≠ 0. However, these functions are regularly varying.

Notes

References

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• .

References

1. See {{harv. Galambos. Seneta. 1973
2. See {{harv. Bingham. Goldie. Teugels. 1987.