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 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.^[1]^[2]

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:^[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 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

^ ^a ^b ^c See (Galambos & Seneta 1973)
^ ^a ^b 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] See (Galambos & Seneta 1973)

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

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