Slowly varying function

Template:Short description 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,

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

Definition 2. Let L : (0, +∞) → (0, +∞). Then L is a regularly varying function if and only if $\mathrm{\forall }a>0,g_L(a)=\underset{x\to \mathrm{\infty }}{\lim }\frac{L(ax)}{L(x)}\in {\mathbb{ℝ}}^{+}$. 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

Theorem 1. The limit in definitions 1 and 2 is uniform if Template:Mvar 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

• Template:Mvar is a real number,
• Template:Mvar 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 Template:Mvar is called the index of regular variation.

Karamata representation theorem

Theorem 3. A function Template:Mvar 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

$L(x)=\exp (\eta (x)+\underset{B}{\overset{x}{\int }}\frac{\epsilon (t)}{t}dt)$

where

• η(x) is a bounded measurable function of a real variable converging to a finite number as Template:Mvar goes to infinity
• ε(x) is a bounded measurable function of a real variable converging to zero as Template:Mvar goes to infinity.

Examples

• If Template:Mvar is a measurable function and has a limit

$\underset{x\to \mathrm{\infty }}{\lim }L(x)=b\in (0,\mathrm{\infty }),$

then Template:Mvar is a slowly varying function.

• For any β ∈ R, the function L(x) = log^{Template:Hairspβ}Template:Hairspx is slowly varying.
• The function L(x) = x is not slowly varying, nor is L(x) = x^{Template:Hairspβ} 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

• .