Abstract
We consider a random walk {S n} with dependent heavy-tailed increments and negative drift. We study the asymptotics for the tail probability P{sup n S n >x} as x→∞. If the increments of {S n} are independent then the exact asymptotic behavior of P{sup n S n >x} is well known. We investigate the case in which the increments are given as a one-sided asymptotically stationary linear process. The tail behavior of sup n S n turns out to depend heavily on the coefficients of this linear process.
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References
Asmussen S., Henriksen L. Fløe, and Klüppelberg C., “Large claims approximations for risk processes in a Markovian environment,” Stochastic Process. Appl., 54, 29–43 (1994).
Asmussen S. and Højgaard B., “Ruin probability approximations for Markov-modulated risk processes with heavy tails,” Theory Random Proc., 2, 96–107 (1996).
Asmussen S., Schmidli H., and Schmidt V., “Tail probabilities for non-standard risk and queueing processes with subexponential jumps,” Adv. in Appl. Probab., 31, 422–447 (1999).
Baccelli F., Schlegel S., and Schmidt V., “Asymptotics of stochastic networks with subexponential service times,” Queueing Systems Theory Appl., 33, 205–232 (1999).
Jelenkovi? P. R. and Lazar A. A., “A network multiplexer with multiple time scale and subexponential arrivals,” in: Stochastic Networks: Stability and Rare Events, Springer, New York, 1996, pp. 215–235.
Mikosch T. and Samorodnitsky G., “The supremum of a negative drift random walk with dependent heavy-tailed steps,” Ann. Appl. Probab., 10, 1025–1064 (2000).
Embrechts P. and Veraverbeke N., “Estimates for the probability of ruin with special emphasis on the possibility of large claims,” Insurance Math. Econom., 1, 55–72 (1982).
Asmussen S., Ruin Probabilities, World Sci. Publ. Co., Singapore (2000).
Embrechts P., Klüppelberg C., and Mikosch T., Modelling Extremal Events for Insurance and Finance, Springer-Verlag, Berlin (1997).
Rolski T., Schmidli H., Schmidt V., and Teugels J., Stochastic Processes for Insurance and Finance, John Wiley & Sons, Chichester (1999).
Korshunov D., “On the distribution tail of the maxima of a random walk,” Stochastic Process. Appl., 72, 97–103 (1997).
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Korshunov, D.A., Schlegel, S. & Schmidt, V. Asymptotics for Random Walks with Dependent Heavy-Tailed Increments. Siberian Mathematical Journal 44, 833–844 (2003). https://doi.org/10.1023/A:1025940920770
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DOI: https://doi.org/10.1023/A:1025940920770