Large deviations for some logarithmic means in the case of random variables with thin tails

In this paper we generalize Theorem 3.3 in Giuliano and Macci (2011). More precisely we prove the full large deviation principle without assuming a particular condition in that theorem and, moreover, we give some results for the case of random variables with thin tails (and not super-exponential tails). As an application we deduce some consequences for the logarithmic means of some random variables expressed in terms of a C-process.