en>Magioladitis: clean up using AWB (10058)

2014-03-28T06:44:06Z

clean up using AWB (10058)

en>Bomazi: Disambiguated: form → Homogeneous polynomial using Dab solver

2011-03-28T09:13:09Z

Disambiguated: form → Homogeneous polynomial using Dab solver

New page

In [[real analysis]], a branch of [[mathematics]], a '''slowly varying function''' is a function resembling a function converging at infinity. A '''regularly varying function''' resembles a [[power law]] function near infinity. Slowly varying and regularly varying functions are important in [[probability theory]].

== Definition ==

A function ''L'': (0,∞) → (0, ∞) is called ''slowly varying'' (at infinity) if for all ''a'' > 0,
:<math>\lim_{x \to \infty} \frac{L(ax)}{L(x)}=1.</math>

If the limit
:<math> g(a) = \lim_{x \to \infty} \frac{L(ax)}{L(x)}</math>
is finite but nonzero for every ''a'' > 0, the function ''L'' is called a ''regularly varying function''.

These definitions are due to [[Jovan Karamata]] {{harv|Galambos|Seneta|1973}}.
Regular variation is the subject of {{harv|Bingham|Goldie|Teugels|1989}}

== Examples ==
* If ''L'' has a limit
::<math>\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, neither is ''L''(''x'')=''x''''β'' for any real ''β'';≠0. However, they are regularly varying.

== Properties ==

Some important properties are {{harv|Galambos|Seneta|1973}}:
* The limit in the definition is [[uniform convergence|uniform]] if ''a'' is restricted to a finite interval.
* ''Karamata's characterization theorem'': every regularly varying function is of the form ''x'' ''β''''L''(''x'') where ''β'' ≥ 0 and ''L'' is a slowly varying function. That is, the function ''g''(''a'') in the definition has to be of the form ''a''''ρ''; the number ''ρ'' is called the index of regular variation.
* ''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

::<math> L(x) = \exp \left( \eta(x) + \int_B^x \frac{\varepsilon(t)}{t} \,dt \right) </math>

:where ''η''(''x'') converges to a finite number and ''ε''(''x'') converges to zero as ''x'' goes to infinity, and both functions are measurable and bounded.

==References==

* {{eom|id=k/k110030|first=N.H.|last=Bingham}}

* {{Citation | last1=Bingham | first1=N.H. | last2=Goldie | first2=C.M. | last3=Teugels | first3=J.L. | title=Regular Variation | year=1989 | publisher= Cambridge University Press}}.

* {{Citation | 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}}.

[[Category:Real analysis]]
[[Category:Tauberian theorems]]
[[Category:Types of functions]]

Polynomial SOS - Revision history

en>Magioladitis: clean up using AWB (10058)

en>Bomazi: Disambiguated: form → Homogeneous polynomial using Dab solver