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Power estimates for ruin probabilities

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Harri Nyrhinen
Harri Nyrhinen*
Affiliation:
University of Helsinki
*
∗ Postal address: Department of Mathematics and Statistics, University of Helsinki, PO Box 68, Helsinki, FIN-00014, Finland. Email address: harri.nyrhinen@helsinki.fi
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Abstract

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Let X1, X2,… be real-valued random variables. For u>0, define the time of ruin T = T(u) by T = inf{n: X1+⋯+Xn>u} or T=∞ if X1+⋯+Xn≤u for every n = 1,2,…. We are interested in the ruin probabilities of general processes {Xn} for large u. In the presence of heavy tails, one often finds power estimates. Our objective is to specify the associated powers and provide the crude estimate P(T≤xu)≈u−R(x) for large u, for a given x∈ℝ. The rate R(x) will be described by means of tails of partial sums and maxima of {Xn}. We also extend our results to the case of the infinite time horizon.

Keywords

Ruin probabilityheavy tailconditional independencelarge deviation

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Secondary: 60F10: Large deviations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 3 , September 2005 , pp. 726 - 742
Copyright
Copyright © Applied Probability Trust 2005 

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