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,[1][2] and have found several important applications, for example in
probability theory.
Basic definitions
Definition 1. A
measurable functionL : (0, +∞) → (0, +∞) is called slowly varying (at infinity) if for all a > 0,
Definition 2. Let L : (0, +∞) → (0, +∞). Then L is a regularly varying function if and only if . In particular, the
limit must be finite.
Note. In the regularly varying case, the sum of two slowly varying functions is again slowly varying function.[clarification needed]
Basic properties
Regularly varying functions have some important properties:[1] 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).