Large deviations for a class of tempered subordinators and their inverse processes

We consider a class of tempered subordinators, namely a class of subordinators with one-dimensional marginal tempered distributions which belong to a family studied in [3]. The main contribution in this paper concerns a non-central moderate deviations result. In fact we consider a family of non-Gaussian equally distributed random variables (so they are trivially weakly convergent), some other related random variables that converge to a constant (and this follows from a large deviation principle), and a class of large deviation principles that fill the gap between these two asymptotic regimes. Some other minor results concern large deviations for the inverse of the tempered subordinators considered in this paper; actually, in some results, these inverse processes appear as random time-changes of other independent processes.