On an Extension of the Concept of Slowly Varying Function with Applications to Large Deviation Limit Theorems

TL;DR

This paper extends Karamata's integral representation to a broader class of functions called $$-locally constant functions and applies this to improve large deviation results for sums of random variables with heavy tails.

Contribution

It introduces $$-locally constant functions, generalizing slowly varying functions, and applies this to extend large deviation theorems for heavy-tailed distributions.

Findings

Extended large deviation results for heavy-tailed sums.

Broader class of functions for asymptotic analysis.

Improved understanding of tail behavior in probability.

Abstract

Karamata's integral representation for slowly varying functions is extended to a broader class of the so-called $\psi$-locally constant functions, i.e. functions $f(x)>0$ having the property that, for a given non-decreasing function $\psi (x)$ and any fixed $v$, $f (x+v\psi(x))/f(x) \to 1 $ as $x\to\infty$. We consider applications of such functions to extending known results on large deviations of sums of random variables with regularly varying distribution tails.