Slowly varying function

Function in mathematics

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, +∞) → (0, +∞) is called slowly varying (at infinity) if for all a > 0,

lim x → ∞ L ( a x ) L ( x ) = 1. {\displaystyle \lim _{x\to \infty }{\frac {L(ax)}{L(x)}}=1.}

Definition 2. Let L : (0, +∞) → (0, +∞). Then L is a regularly varying function if and only if ∀ a > 0 , g L ( a ) = lim x → ∞ L ( a x ) L ( x ) ∈ R + {\displaystyle \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, +∞) → (0, +∞) is of the form

f ( x ) = x β L ( x ) {\displaystyle 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 ρ {\displaystyle 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 ≥ B the function can be written in the form

L ( x ) = exp ⁡ ( η ( x ) + ∫ B x ε ( t ) t d t ) {\displaystyle 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 → ∞ L ( x ) = b ∈ ( 0 , ∞ ) , {\displaystyle \lim _{x\to \infty }L(x)=b\in (0,\infty ),}
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, nor is L(x) = x β for any real β ≠ 0. However, these functions are regularly varying.

See also

Notes

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.