### Abstract

We derive an expression in terms of the Wright function for the density of the ﬁrst-passage times (or FPT’s) for the Poisson-Erlang Levy processes. For Poisson-exponential processes, we establish an analogue of Zolotarev space-time duality between the original process and its FPT process “truncated” at zero. We show that an asymptotic duality holds in the sense of weak convergence, thereby providing an interpretation of the Letac-Mora reciprocity. The corresponding limits in the sense of convergence in mean
and in mean square have an additional multiplier, which is also present in the asymptotic relationship between the marginals of a Poisson-Erlang process and its “truncated” FPT process. We prove that for Poisson-exponential processes, the FPT and the overshoot are independent.

Original language | English |
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Pages (from-to) | 283-302 |

Number of pages | 20 |

Journal | Communications on Stochastic Analysis |

Volume | 7 |

Issue number | 2 |

Publication status | Published - Jun 2013 |

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## Cite this

Paris, R. B., & Vinogradov, V. (2013). Fluctuation properties of compound Poisson-Erlang Lévy processes.

*Communications on Stochastic Analysis*,*7*(2), 283-302. https://www.math.lsu.edu/cosa/7-2-08[387].pdf