  Renewal processes with heavy-tailed power law distributed sojourn times are
commonly encountered in physical modelling and so typical fluctuations of
observables of interest have been investigated in detail. To describe rare
events the rate function approach from large deviation theory does not hold and
new tools must be considered. Here we investigate the large deviations of the
number of renewals, the forward and backward recurrence time, the occupation
time, and the time interval straddling the observation time. We show how
non-normalized densities describe these rare fluctuations, and how moments of
certain observables are obtained from these limiting laws. Numerical
simulations illustrate our results showing the deviations from arcsine, Dynkin,
Darling-Kac, L{\'e}vy and Lamperti laws.
