On the upper bound in Varadhan’s Lemma

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, in this paper 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. Keywords. Varadhan's Lemma exponential integrals large deviations principle upper bound
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H.M. Jansen, Michel Mandjes, Koen de Turck, S Wittevrongel. On the upper bound in Varadhan’s Lemma. Statistics and Probability Letters, Elsevier, 2015, 103, pp.24-29. ⟨10.1016/j.spl.2015.04.005⟩. ⟨hal-01327058⟩

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