Slowly varying function

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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[edit]

A 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.

If the limit

 g(a) = \lim_{x \to \infty} \frac{L(ax)}{L(x)}

is finite but nonzero for every a > 0, the function L is called a regularly varying function.

These definitions are due to Jovan Karamata (Galambos & Seneta 1973). Regular variation is the subject of (Bingham, Goldie & Teugels 1989)

Examples[edit]

  • If L has a limit
\lim_{x \to \infty} L(x) = b \in (0,\infty),
then L is a slowly varying function.
  • For any βR, the function L(x)= logβ x is slowly varying.
  • The function L(x)=x is not slowly varying, neither is L(x)=xβ for any real β;≠0. However, they are regularly varying.

Properties[edit]

Some important properties are (Galambos & Seneta 1973):

  • The limit in the definition is uniform if a is restricted to a finite interval.
  • Karamata's characterization theorem: every regularly varying function is of the form x βL(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 aρ; 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 xB 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) converges to a finite number and ε(x) converges to zero as x goes to infinity, and both functions are measurable and bounded.

References[edit]

  • Bingham, N.H.; Goldie, C.M.; Teugels, J.L. (1989), Regular Variation, Cambridge University Press .