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,See {{harv|Galambos|Seneta|1973}}See {{harv|Bingham|Goldie|Teugels|1987}}. and have found several important applications, for example in probability theory.

Basic definitions

{{EquationRef|1|Definition 1}}. A measurable function {{math|L : (0, +∞) → (0, +∞)}} is called slowly varying (at infinity) if for all {{math|a > 0}},

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

{{EquationRef|2|Definition 2}}. Let {{math|L : (0, +∞) → (0, +∞)}}. Then {{math|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 {{harvtxt|Bingham|Goldie|Teugels|1987}}.

=Uniformity of the limiting behaviour=

{{EquationRef|3|Theorem 1}}. The limit in {{EquationNote|1|definitions 1}} and {{EquationNote|2|2}} is uniform if {{mvar|a}} is restricted to a compact interval.

=Karamata's characterization theorem=

{{EquationRef|4|Theorem 2}}. Every regularly varying function {{math|f : (0, +∞) → (0, +∞)}} is of the form

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

where

{{mvar|β}} is a real number,
{{mvar|L}} is a slowly varying function.

Note. This implies that the function {{math|g(a)}} in {{EquationNote|2|definition 2}} has necessarily to be of the following form

: $g(a)=a^\rho$

where the real number {{mvar|ρ}} is called the index of regular variation.

=Karamata representation theorem=

{{EquationRef|5|Theorem 3}}. A function {{mvar|L}} is slowly varying if and only if there exists {{math|B > 0}} such that for all {{math|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

{{math|η(x)}} is a bounded measurable function of a real variable converging to a finite number as {{mvar|x}} goes to infinity
{{math|ε(x)}} is a bounded measurable function of a real variable converging to zero as {{mvar|x}} goes to infinity.

Examples

If {{mvar|L}} is a measurable function and has a limit

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

:then {{mvar|L}} is a slowly varying function.

For any {{math|β ∈ R}}, the function {{math|L(x) {{=}} log^{{hairsp}}β{{hairsp}}x}} is slowly varying.
The function {{math|L(x) {{=}} x}} is not slowly varying, nor is {{math|L(x) {{=}} x^{{hairsp}}β}} for any real {{math|β ≠ 0}}. However, these functions are regularly varying.

Notes

References

{{SpringerEOM|title=Karamata theory|oldid=25937|first=N.H.|last=Bingham}}
{{Citation

| last=Bingham

| first=N. H.

| last2=Goldie

| first2=C. M.

| last3=Teugels

| first3=J. L.

| title=Regular Variation

| place=Cambridge

| publisher=Cambridge University Press

| series=Encyclopedia of Mathematics and its Applications

| volume=27

| year=1987

| edition=

| url=https://archive.org/details/regularvariation0000bing

| doi=

| isbn=0-521-30787-2

| mr=0898871

| zbl=0617.26001

| url-access=registration

}}

{{Citation |author2-link=Eugene Seneta | last1=Galambos | first1=J. | last2=Seneta | first2=E. | title=Regularly Varying Sequences | year=1973 | journal=Proceedings of the American Mathematical Society | issn=0002-9939 | volume=41 | issue=1 | pages=110–116 | doi=10.2307/2038824 | jstor=2038824| doi-access=free }}.

Category:Real analysis

Category:Tauberian theorems

Category:Types of functions