Regularly Varying Function

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



References

2015

[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

[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

(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]