Research output: Contribution to journal › Article

Aleksandar Mijatovic, Martijn Pistorius

Original language | English |
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Article number | 43 |

Pages (from-to) | 1-18 |

Number of pages | 18 |

Journal | Electronic Communications in Probability |

Volume | 21 |

DOIs | |

Accepted/In press | 9 May 2016 |

Published | 23 May 2016 |

Additional links |

**Joint asymptotic distribution_MIJATOVIC_Accepted 09May2016_Gold VoR**Joint_asymptotic_distribution_MIJATOVIC_Accepted_09May2016_Gold_VoR.pdf, 350 KB, application/pdf

Uploaded date:14 Nov 2016

Version:Final published version

Licence:CC BY

Let τ(x) be the first time that the reflected process Y of a Lévy process X crosses x>0. The main aim of this paper is to investigate the joint asymptotic distribution of Y(t)=X(t)−inf_{0≤s≤t}X(s) and the path functionals Z(x)=Y(τ(x))−x and m(t)=sup_{0≤s≤t}Y(s)−y*(t) for a certain non-linear curve y*(t). We restrict to Lévy processes X satisfying Cramér’s condition, a non-lattice condition and the moment conditions that E[|X(1)|] and E[exp(γX(1))|X(1)|] are finite (where γ denotes the Cramér coefficient). We prove that Y(t) and Z(x) are asymptotically independent as min{t,x}→∞ and characterise the law of the limit (Y∞,Z∞). Moreover, if y*(t)=γ^{−1}log(t) and min{t,x}→∞ in such a way that texp{−γx}→0, then we show that Y(t), Z(x) and m(t) are asymptotically independent and derive the explicit form of the joint weak limit (Y∞,Z∞,m∞). The proof is based on excursion theory, Theorem 1 in [7] and our characterisation of the law (Y∞,Z∞).

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