Regularly Varying Function

A Regularly Varying Function is a real variable function that behaves, at the infinity, similarly to a power law function near the infinity.

• Context:
  • A real variable function, [math]\displaystyle{ f(x) }[/math] is regularly varying function if [math]\displaystyle{ \lim_{x \to \infty} \frac{f(ax)}{f(x)} = a^\rho }[/math] is finite and nonzero for every [math]\displaystyle{ a \gt 0 }[/math]. Therefore, [math]\displaystyle{ \rho }[/math] is a non negative real number and it is usually called the index of regular variation.
• Example(s):
  • A Polynomial Function near the infinity.
• Counter-Example(s):
  • ...
• See: Slowly Varying Function, Power Law.

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References

2015

• (Wikipedia, 2015) ⇒ https://www.wikipedia.com/en/Slowly_varying_function
  • QUOTE: A function L : (0,+∞) → (0,+∞) for which the limit

[math]\displaystyle{ g(a) = \lim_{x \to \infty} \frac{L(ax)}{L(x)} }[/math]

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

2015

• (Amsaleg et al., 2015) ⇒ Laurent Amsaleg, Oussama Chelly, Teddy Furon, Stéphane Girard, Michael E. Houle, Ken-ichi Kawarabayashi, and Michael Nett. (2015). “Estimating Local Intrinsic Dimensionality.” In: Proceedings of the 21st ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD-2015). ISBN:978-1-4503-3664-2 doi:10.1145/2783258.2783405
  • QUOTE: The Fisher-Tippett-Gnedenko Theorem and the PickandsBalkema-de Haan Theorem have been shown to be equivalent to a third characterization of the tail behavior, in terms of regularly-varying (RV) functions. The asymptotic cumulative distribution of [math]\displaystyle{ X }[/math] in the tail [math]\displaystyle{ [0, w] }[/math] can be expressed as [math]\displaystyle{ F_X(x) = x^\kappa\ell_X(1/x) }[/math] where [math]\displaystyle{ \ell_X }[/math] is differentiable and slowly varying; that is, for all [math]\displaystyle{ c \gt 0; \ell_X }[/math] satisfies

[math]\displaystyle{ \lim_{x \to \infty}\frac{\ell_X(ct)}{\ell_X(t)} = 1 }[/math]

.[math]\displaystyle{ F_X }[/math] restricted to [0, w) is itself said to be regularly varying with index [math]\displaystyle{ \kappa }[/math]. In particular, a cumulative distribution [math]\displaystyle{ F ∈ F_{GEV} }[/math] has [math]\displaystyle{ ξ \lt 0 }[/math] if and only if F is RV and has a finite endpoint. Note that the slowly-varying component [math]\displaystyle{ \ell_X(1/x) }[/math] of [math]\displaystyle{ F_X }[/math] is not necessarily constant as x tends to zero.

1989

• (Bingham et al.,1989) ⇒ author: NicholasH. Bingham, Charles M. Goldie, and Jef L. Teugels. (1989). “Regular variation." in Vol. 27. Cambridge university press.
  • QUOTE: Theorem 1.5.6 (Potter's Theorem)

(i) If [math]\displaystyle{ \ell }[/math] is slowly varying then for any chosen constants [math]\displaystyle{ A \gt 1 \quad \delta \gt 0 }[/math] there exists [math]\displaystyle{ X = X(A, \delta) }[/math] such that :

[math]\displaystyle{ \ell(y)/\ell(x)\leq A\; max\{(y/x)^\delta, (y/x)^{-\delta}\} (x\geq X, y\geq X) }[/math]

(ii) If, further, [math]\displaystyle{ \ell }[/math] is bounded away from 0 and [math]\displaystyle{ \infty }[/math] on every compact subset of [math]\displaystyle{ [0,\infty] }[/math], then for every [math]\displaystyle{ b\gt \delta }[/math] there exists [math]\displaystyle{ A'= A'(\delta)\gt 1 }[/math] such that :

[math]\displaystyle{ \ell(y)/\ell(x) \lt A' \;max\{(y/x)^\delta (y/x)^{-delta}\} (x \gt 0, y\gt 0) }[/math]

(iii) If [math]\displaystyle{ f }[/math]is regularly varying of index [math]\displaystyle{ \rho }[/math] then for any chosen [math]\displaystyle{ A\gt 1, \;delta\gt 0 }[/math] there exists [math]\displaystyle{ X=X(A,\delta) }[/math] such that:

[math]\displaystyle{ f(y)/f(x)\leq A \;max\{(y/x)^{\rho+\delta},(y/x)^{\rho-\delta}\} (x\geq X ,y\geq X) }[/math]

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