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Asymptotic behaviour of first passage time distributions for subordinators. (English) Zbl 1333.60094
Summary: 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 N. C. Jain and W. E. Pruitt [Ann. Probab. 15, 75–101 (1987; Zbl 0617.60023)]. 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 in [Probab. Theory Relat. Fields 157, No. 1–2, 1–45 (2013; Zbl 1286.60042)] for non-monotone Levy processes. For subordinators, an approach different to that in [Doney and Rivero, loc. cit.] 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.

MSC:
60G51 Processes with independent increments; Lévy processes
60F10 Large deviations
62E20 Asymptotic distribution theory in statistics
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