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.

. A function $L : (0,+\infty) \to (0,+\infty)$ is called slowly varying (at infinity) if for all $a > 0$ ,

. A function $L : (0,+\infty) \to (0,+\infty)$ for which the limit

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

These definitions are due to Jovan Karamata.

Regularly varying functions have some important properties: 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).

. The limit in definitions 1 and 2 is uniform if $a$ is restricted to a compact interval.

. Every regularly varying function $f$ is of the form

where

Note. This implies that the function $g (a)$ in definition 2 has necessarily to be of the following form

where the real number $ρ$ is called the index of regular variation.

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