Slowly varying function

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, and have found several important applications, for example in probability theory and extreme value 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.