Slowly varying function

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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[edit]

Definition 1. 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.

Definition 2. A function L : (0,+∞) → (0,+∞) for which the limit

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

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

These definitions are due to Jovan Karamata.[1][2]

Basic properties[edit]

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[edit]

Theorem 1. The limit in definitions 1 and 2 is uniform if a is restricted to a finite interval.

Karamata's characterization theorem[edit]

Theorem 2. Every regularly varying function f is of the form

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

where

  • β ≥ 0 is a non negative 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 non negative real number ρ is called the index of regular variation.

Karamata representation theorem[edit]

Theorem 3. 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

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, these functions are regularly varying.

See also[edit]

Notes[edit]

References[edit]