Subexponential asymptotics for stochastic processes: Extremal behavior, stationary distributions and first passage probabilities

Annals of Applied Probability

Volumn 8, Issue 2, 1998, Pages 354-374

  Asmussen, Søren ^a  

^a LUND UNIVERSITY   (Sweden)

Author keywords

Cycle maximum; Extremal index; Extreme values; Frechet distribution; Gumbel distribution; Interest force; Level crossings; Maximum domain of attraction; Overshoot distribution; Random walk; Rare event; Regular variation; Ruin probability; Stable process

Indexed keywords

EID: 0032221191     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/aoap/1028903531     Document Type: Article

References (30)

1. ANANTHARAM, V. (1988). How large delays build up in a GI/GI/1 queue. Queueing Systems Theory Appl. 5 345-368.
2. ASMUSSEN, S. (1982). Conditioned limit theorems relating a random walk to its associate, with applications to risk reserve processes and the GI/G/1 queue. Adv. in Appl. Probab. 14 143-170.
3. ASMUSSEN, S. (1987). Applied Probability and Queues. Wiley, New York.
4. ASMUSSEN, S. (1996). Rare events in the presence of heavy tails. In Stochastic Networks: Rare Events and Stability (P. Glasserman, K. Sigman and D. Yao, eds.) 197-214. Springer, New York.
5. ASMUSSEN, S. and KLÜPPELBEBG, C. (1995). Large deviations results in the presence of heavy tails, with applications to insurance risk. Stochastic Process. Appl. 64 103-125.
6. ASMUSSEN, S. and KLÜPPELBERG, C. (1997). Stationary M/G/1 excursions in the presence of heavy tails. J. Appl. Probab. 34 208-212.
7. ASMUSSEN, S. and NIELSEN, H. M. (1995). Ruin probabilities via local adjustment coefficients. J. Appl. Probab. 32 736-755.
8. ASMUSSEN, S. and SCHOCK PETERSEN, S. (1989). Ruin probabilities expressed in terms of storage processes. Adv. in Appl. Probab. 20 913-916.
9. BALKEMA, A. A. and DE HAAN, L. (1974). Residual life-time at great age. Ann. Probab. 2 792-804.
10. HERMAN, S. M. (1962). Limit distribution of the maximum term in a sequence of dependent random variables. Ann. Math. Statist. 33 894-908.
11. BROCKWELL, P. J., RESNICK, S. I. and TWEEDIE, R. L. (1982). Storage processes with general release rule and additive inputs. Adv. in Appl. Probab. 14 392-433.
12. DASSIOS, A. and EMBRECHTS, P. (1989). Martingales and insurance risk. Stochastic Models 5 181-217.
13. DURRETT, R. (1980). Conditioned limit theorems for random walks with negative drift. Z. Wahrsch. Verw. Gebiete 52 277-287.
14. EMBRECHTS, P. and GOLDIE, C. M. (1980). On closure and factorization properties of subexponential and related distributions. J. Austral. Math. Soc. Ser. A 29 243-256.
15. EMBRECHTS, P., KLÜPPELBERG, C. and MIKOSCH, T. (1997). Extremal Events in Finance and Insurance. Springer, New York.
16. EMBRECHTS, P. and VERAVERBEKE, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom. 1 55-72.
17. GELUB, J. L. and DE HAAN, L. (1987). Regular Variation, Extensions and Tauberian Theorems. CWI Tract 40. CWI, Amsterdam.
18. GNEDENKO, B. V. and KOVALENKO, I. N. (1989). Introduction to Queueing Theory, 2nd ed. Birkhäuser, Basel.
19. GOLDIE, C. and RESNICK, S. I. (1988). Distributions that are both subexponential and in the domain of attraction of an extreme-value distribution. Adv. in Appl. Probab. 20 706-718.
20. HARRISON, J. M. and RESNICK, S. I. (1976). The stationary distribution and first exit probabilities of a storage process with general release rule. Math. Oper. Res. 1 347-358.
21. HARRISON, J. M. and RESNICK, S. I. (1977). The recurrence classification of risk and storage processes. Math. Oper. Res. 3 57-66.
22. IGLEHART, D. L. (1972). Extreme values in the GI/G/1 queue. Ann. Math. Statist. 43 627-635.
23. KEILSON, J. (1979). Markov Chain Models-Rarity and Exponentiality. Springer, New York.
24. KLÜPPELBERG, C. (1988). Subexponential distributions and integrated tails. J. Appl. Probab. 25 132-141.
25. KLÜPPELBERG, C. and STADTMÜLLER, U. (1995). Ruin probabilities in the presence of heavytails and interest rates. Scared. Actuar. J. 49-58.
26. LEADBETTER, M. R., LINDGREN, G. and ROOTZÉN, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, New York.
27. RESNICK, S. I. (1987). Extreme Values, Regular Variation Point and Processes. Springer, New York.
28. ROOTZÉN, H. (1988). Maxima and exceedances of stationary Markov chains. Adv. in Appl. Probab. 20 371-390.
29. SUNDT, B. and TEUGELS, J. L. (1995). Ruin estimates under interest force. Insurance Math. Econom. 16 7-22.
30. SUNDT, B. and TEUGELS, J. L. (1996). The adjustment coefficient in ruin estimates under interest force. Insurance Math. Econom.