  This paper deals with the large deviations behavior of a stochastic process
called thinned Levy process. This process appeared recently as a
stochastic-process limit in the context of critical inhomogeneous random
graphs. The process has a strong negative drift, while we are interested in the
rare event of the process being positive at large times. To characterize this
rare event, we identify a tilted measure. This presents some challenges
inherent to the power-law nature of the thinned Levy process. General
principles prescribe that the tilt should follow from a variational problem,
but in the case of the thinned Levy process this involves a Riemann sum that is
hard to control. We choose to approximate the Riemann sum by its limiting
integral, derive the first-order correction term, and prove that the tilt that
follows from the corresponding approximate variational problem is sufficient to
establish the large deviations results.
