The probability of exceeding a High boundary on a random time interval for a heavy-tailed random walk

Annals of Applied Probability

Volumn 15, Issue 3, 2005, Pages 1936-1957

  Foss, Serguei ^a   Palmowski, Zbigniew ^a,b,c   Zachary, Stan ^a  

^a HERIOT WATT UNIVERSITY   (United Kingdom)

^b UNIVERSITY OF WROC AW   (Poland)

^c UTRECHT UNIVERSITY   (Netherlands)

Author keywords

Boundary; Random walk; Ruin probability; Subexponential distributions

Indexed keywords

EID: 23944442984     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/105051605000000269     Document Type: Review

References (24)

1. ALSMEYER, G. and IRLE, A. (1986). Asymptotic expansions for the variance of stopping times in nonlinear renewal theory. Stochastic Process. Appl. 23 235-258.
2. ASMUSSEN, S. (1998). Subexponential asymptotics for stochastic processes: Extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Probab. 8 354-374.
3. ASMUSSEN, S. (2003). Applied Probability and Queues, 2nd ed. Springer, New York.
4. BACCELLI, F. and Foss, S. (2004). Moments and tails in monotone-separable stochastic networks. Ann. Appl. Probab. 14 612-650.
5. BOROVKOV, A. A. (2000). Estimates for the distribution of sums and maxima of sums of random variables without the Cramer condition. Siberian Math. J. 41 811-848.
6. BOROVKOV, A. A. and BOROVKOV, K. A. (2001). On large deviations probabilities for random walks. I. Regularly varying distribution tails. Theory Probab. Appl. 46 193-213.
7. BOROVKOV, A. A. and Foss, S. (1999). Estimates of excess over an arbitrary boundary for a random walk and their applications. Theory Probab. Appl. 44 249-277.
8. BRAVERMAN, M., MIKOSCH, T. and SAMORODNITSKY, G. (2002). Tail probabilities of subadditive functionals of Lévy processes. Ann. Appl. Probab. 12 69-100.
9. DENISOV, D. (2004). A short proof of Asmussen's theorem. Markov Process. Related Fields. To appear.
10. DENISOV, D., FOSS, S. and KORSHUNOV, D. (2004). Tail asymptotics for the supremum of a random walk when the mean is not finite. Queueing Systems 46 5-33.
11. EMBRECHTS, P., KLÜPPELBERG, C. and MIKOSCH, T. (1997). Modelling Extremal Events. Springer, Berlin.
12. EMBRECHTS, P. and VERAVERBEKE, N. (1982). Estimates for the probability of ruin with special emphasis on the probability of large claims. Insurance Math. Econom. 1 55-72.
13. FELLER, W. (1971). An Introduction to Probability Theory and Its Applications. Wiley, New York.
14. Foss, S. and ZACHARY, S. (2003). The maximum on a random time interval of a random walk with long-tailed increments and negative drift. Ann. Appl. Probab. 13 37-53.
15. HEATH, D., RESNICK, S. and SAMORODNITSKY, G. (1997). Patterns of buffer overflow in a class of queues with long memory in the input stream. Ann. Appl. Probab. 7 1021-1057.
16. KLÜPPELBERG, C. (1988). Subexponential distributions and integrated tails. J. Appl. Probab. 35 325-347.
17. KORSHUNOV, D. (2002). Large-deviation probabilities for maxima of sums of independent random variables with negative mean and Subexponential distribution. Theory Probab. Appl. 46 355-366.
18. TANG, Q. (2004). Uniform estimates for the tail probability of maxima over finite horizons with subexponential tails. Probab. Engrg. Inform. Sci. 18 71-86.
19. VERAVERBEKE, N. (1977). Asymptotic behavior of Wiener-Hopf factors of a random walk. Stochastic Process. Appl. 5 27-37.
20. WALK, H. (1989). Nonlinear renewal theory under growth conditions. Stochastic Process. Appl. 32 289-303.
21. WOODROOFE, M. (1976). A renewal theorem for curved boundaries and moments of first passage times. Ann. Probab. 4 67-80.
22. WOODROOFE, M. (1982). Nonlinear Renewal Theory in Sequential Analysis. SIAM, Philadelphia, PA.
23. ZACHARY, S. (2004). A note on Veraverbeke's theorem. Queueing Systems 46 9-14.
24. ZHANG, C. H. (1988). A nonlinear renewal theory. Ann. Probab. 16 793-824.