In this paper we establish local estimates for the first passage time of a subordinator under the assumption that it belongs to the Feller class, either at zero or infinity, having as a particular case the subordinators which are in the domain of attraction of a stable distribution, either at zero or infinity. To derive these results we first obtain uniform local estimates for the one dimensional distribution of such a subordinator, which sharpen those obtained by Jain and Pruitt. In the particular case of a subordinator in the domain of attraction of a stable distribution our results are the analogue of the results obtained by the authors for non-monotone Levy processes. For subordinators an approach different to that in  is necessary because the excursion techniques are not available and also because typically in the non-monotone case the tail distribution of the first passage time has polynomial decrease, while in the subordinator case it is exponential.
"Asymptotic behaviour of first passage time distributions for subordinators." Electron. J. Probab. 20 1 - 28, 2015. https://doi.org/10.1214/EJP.v20-3879