  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 is a
non-central moderate deviations result. More precisely we mean a class of large
deviation principles that fill the gap between the (trivial) weak convergence
of some non-Gaussian identically distributed random variables to their common
law, and the convergence of some other related random variables to a constant.
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.
