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Some asymptotic results for transient random walks with applications to insurance risk

Part of: Stochastic processes Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Aleksandras Baltrūnas
Aleksandras Baltrūnas*
Affiliation:
Institute of Mathematics and Informatics, Vilnius
*
∗ Postal address: Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Email address: jurgita@ktl.mii.lt
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Abstract

We consider a real-valued random walk which drifts to -∞ and is such that the step distribution is heavy tailed, say, subexponential. We investigate the asymptotic tail behaviour of the distribution of the upwards first passage times. As an application, we obtain the exact rate of convergence for the ruin probability in finite time. Our result supplements similar theorems in risk theory.

Keywords

transient random walkssubexponential distributionsprecise large deviationsruin probability

MSC classification

Primary: 60G50: Sums of independent random variables; random walks 60K25: Queueing theory
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60G51: Processes with independent increments; L'evy processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 108 - 121
Copyright
Copyright © by the Applied Probability Trust 2001 

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References

Andersen, E. S. (1957). On the collective theory of risk in case of contagion between claims. In Trans. XVth Int. Congress of Actuaries, New York, Vol. II, 219–229.Google Scholar
Asmussen, S., Klüppelberg, C., and Sigman, K. (1999). Sampling at subexponential times, with queueing applications. Stoch. Proc. Appl. 79, 265–286.Google Scholar
Asmussen, S., and Teugels, J. L. (1996). Convergence rates for M/G/1 queues and ruin problems with heavy tails. J. Appl. Prob. 33, 1181–1190.Google Scholar
Baltrūnas, A. (1995). On the asymptotics of one-sided large deviation probabilities. Lithuanian Math. J. 35, 11–17.CrossRefGoogle Scholar
Bertoin, J., and Doney, R. A. (1996). Some asymptotic results for transient random walks. Adv. Appl. Prob. 28, 207–226.CrossRefGoogle Scholar
Chover, J., Ney, P., and Wainger, S. (1973). Functions of probability measures. J. Anal. Math. 26, 255–302.Google Scholar
Doney, R. A. (1989). On the asymptotic behaviour of first passage times for transient random walks. Prob. Theory Rel. Fields 81, 239–246.CrossRefGoogle Scholar
Embrechts, P., and Hawkes, J. (1982). A limit theorem for the tails of discrete infinitely divisible laws with applications to fluctuation theory. J. Austral. Math. Soc. A 32, 412–422.Google Scholar
Embrechts, P., and Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance: Math. Econ. 1, 55–72.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132–141.CrossRefGoogle Scholar
Klüppelberg, C. (1989). Estimation of ruin probabilities by means of hazard rates. Insurance: Math. Econ. 8, 279–285.Google Scholar
Nagaev, A. V. (1977). On a property of sums of independent random variables. Theory Prob. Appl. 22, 335–346.Google Scholar