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.
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Basic definitions
Definition 1. A function L : (0,+∞) → (0,+∞) is called slowly varying (at infinity) if for all a > 0,
Definition 2. A function L : (0,+∞) → (0,+∞) 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.
Basic properties
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).
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 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.
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
where