An extension of the concept of slowly varying function with applications to large deviation limit theorems

Abstract

Karamata's integral representation for slowly varying functions is extended to a broader class of the so-called ψ-locally constant functions, i.e. functions f(x) > 0 having the property that, for a given non-decreasing function ψ(x) and any fixed v, f(x + vψ(x)) / f(x) → 1 as x → ∞. We consider applications of such functions to extending known theorems on large deviations of sums of random variables with regularly varying distribution tails. © Springer-Verlag Berlin Heidelberg 2013.