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August 2017 High order expansions for renewal functions and applications to ruin theory
Dombry Clément, Rabehasaina Landy
Ann. Appl. Probab. 27(4): 2342-2382 (August 2017). DOI: 10.1214/16-AAP1261

Abstract

A high order expansion of the renewal function is provided under the assumption that the inter-renewal time distribution is light tailed with finite moment generating function $g$ on a neighborhood of $0$. This expansion relies on complex analysis and is expressed in terms of the residues of the function $1/(1-g)$. Under the assumption that $g$ can be extended into a meromorphic function on the complex plane and some technical conditions, we obtain even an exact expansion of the renewal function. An application to risk theory is given where we consider high order expansion of the ruin probability for the standard compound Poisson risk model. This precises the well- known Crámer-Lundberg approximation of the ruin probability when the initial reserve is large.

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Dombry Clément. Rabehasaina Landy. "High order expansions for renewal functions and applications to ruin theory." Ann. Appl. Probab. 27 (4) 2342 - 2382, August 2017. https://doi.org/10.1214/16-AAP1261

Information

Received: 1 June 2016; Revised: 1 November 2016; Published: August 2017
First available in Project Euclid: 30 August 2017

zbMATH: 1373.60150
MathSciNet: MR3693528
Digital Object Identifier: 10.1214/16-AAP1261

Subjects:
Primary: 60K05
Secondary: 60K10

Keywords: compound Poisson model , Cramér–Lundberg approximation , Renewal function , Renewal process , ruin probability

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.27 • No. 4 • August 2017
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