On the upper bound in Varadhan's Lemma

TL;DR

This paper extends Varadhan's Lemma by relaxing the conditions on the rate function and integrand, allowing for broader applicability in areas like queueing theory.

Contribution

It generalizes the upper bound in Varadhan's Lemma by weakening assumptions on the rate function and integrand, including cases with non-compact sublevel sets.

Findings

Upper bound holds without compact sublevel sets of the rate function.

Weaker conditions replace upper semicontinuity of the integrand.

Measurability suffices when the rate function is continuous.

Abstract

In this paper, we generalize the upper bound in Varadhan's Lemma. The standard formulation of Varadhan's Lemma contains two important elements, namely an upper semicontinuous integrand and a rate function with compact sublevel sets. However, motivated by results from queueing theory, we do not assume that rate functions have compact sublevel sets. Moreover, we drop the assumption that the integrand is upper semicontinuous and replace it by a weaker condition. We prove that the upper bound in Varadhan's Lemma still holds under these weaker conditions. Additionally, we show that only measurability of the integrand is required when the rate function is continuous.