We consider the process ${x-N(t):tgeq 0}$, where $x>0$ and ${N(t):tgeq 0}$ is a renewal process with light-tailed distributed holding times. We are interested in the joint distribution of $(tau(x),A(x))$ where $tau(x)$ is the first-passage time of ${
x-N(t):tgeq 0}$ to reach zero or a negative value, and $A(x)$ is the corresponding first-passage area. We remark that we can define the sequence ${(tau(n),A(n)):ngeq 1}$ by referring to the concept of integrated random walk. Our aim is to prove asymptotic results as $xtoinfty$ in the fashion of large (and moderate) deviations
In this paper we consider suitable families of power series distributed random variables, and we study their asymptotic behavior in the fashion of large (and moderate) deviations. We also present applications of our results to some fractional counting processes in the literature.
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 resu
lt. 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.
