Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations

Annals of Probability

Volumn 33, Issue 5, 2005, Pages 1992-2035

  Konstantinides, Dimitrios G ^a   Mikosch, Thomas ^a,b,c  

^a UNIVERSITY OF THE AEGEAN   (Greece)

^b UNIVERSITY OF COPENHAGEN   (Denmark)

^c MaPhySto   (Denmark)

Author keywords

Large deviations; Regular variation; Ruin probability; Stochastic recurrence equation

Indexed keywords

EID: 27644474447     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/009117905000000350     Document Type: Article

References (38)

1. BASRAK, B., DAVIS, R. A. and MIKOSCH. T. (2002). Regular variation of GARCH processes. Stochastic Process. Appl. 99 95-116.
2. BAXENDALE, P. H. and KHASMINSKII, R. Z. (1998). Stability index for products of random transformations. Adv. in Appl. Probab. 30 968-988.
3. BILLINGSLEY, P. (1968). Convergence of Probability Measures. Wiley, New York.
4. BINGHAM, N. H., GOLDIE, C. M. and TEUGELS, J. L. (1987). Regular Variation. Cambridge Univ. Press.
5. BOUGEROL, P. and PICARD, N. (1992). Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 1714-1730.
6. BOUSSAMA, F. (1998). Ergodicité, mélange et estimation dans le modelès GARCH. Ph.D. thesis, Univ. Paris 7.
7. BREIMAN, L. (1965). On some limit theorems similar to the arc-sin law. Theory Probab. Appl. 10 323-331.
8. CLINE, D. B. H. and HSING, T. (1991). Large deviation probabilities for sums and maxima of random variables with heavy or subexponential tails. Texas A&M Univ. Preprint.
9. DAVIS, R. A. and HSING, T. (1995). Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Probab. 23 879-917.
10. DAVIS, R. A. and MIKOSCH, T. (1998). The sample autocorrelations of heavy-tailed processes with applications to ARCH. Ann. Statist. 26 2049-2080.
11. DAVIS, R. A. and MIKOSCH, T. (2001). Point process convergence of stochastic volatility processes with application to sample autocorrelations. J. Appl. Probab. Special Volume: A Festschrift for David Vere-Jones 38A 93-104.
12. DAVIS, R. A. and RESNICK, S. I. (1996). Limit theory for bilinear processes with heavy-tailed noise. Ann. Appl. Probab. 6 1191-1210.
13. DOUKHAN, P. (1994). Mixing. Properties and Examples. Lecture Notes in Statist. 85. Springer, New York.
14. DUFRESNE, D. (1990). The distribution of a perpetuity, with application to risk theory. Scand. Actuar. J. 39-79.
15. EMBRECHTS, P., KLÜPPELBERG, C. and MIKOSCH, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin.
16. FELLER, W. (1971). An Introduction to Probability Theory and Its Applications II, 2nd ed. Wiley, New York.
17. GOLDIE, C. M. and GRÜBEL, R. (1996). Perpetuities with thin tails. Adv. in Appl. Probab. 28 463-480.
18. GREY, D. R. (1994). Regular variation in the tail behaviour of solutions to random difference equations. Ann. Appl. Probab. 4 169-183.
19. DE HAAN, L., RESNICK, S. I., ROOTZÉN, H. and DE VRIES, C. (1989). Extremal behaviour of solutions to a stochastic difference equation with applications to ARCH processes. Stochastic Process. Appl. 32 213-224.
20. IBRAGIMOV, I. A. and LINNIK, Yu. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.
21. KESTEN, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207-248.
22. MEYN, S. P. and TWEEDIE, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.
23. MIKOSCH, T. (2003). Modeling dependence and tails of financial time series. In Extreme Values in Finance, Telecommunications, and the Environment (B. Finkenstädt and H. Rootén, eds.) 185-286. Chapman and Hall, Boca Raton.
24. MIKOSCH, T. and NAGAEV, A. V. (1998). Large deviations of heavy-tailed sums with applications to insurance. Extremes 1 81-110.
25. MIKOSCH, T. and SAMORODNITSKY, G. (2000). The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Probab. 10 1025-1064.
26. MIKOSCH, T. and SAMORODNITSKY, G. (2000). Ruin probability with claims modeled by a stationary ergodic stable process. Ann. Probab. 28 1814-1851.
27. MIKOSCH, T. and STRAUMANN, D. (2005). Stable limits of martingale transforms with application to the estimation of GARCH parameters. Ann. Statist. To appear.
28. MOKKADEM, A. (1990). Propriétés de mélange des processus autorégressifs polynomiaux. Ann. Inst. H. Poincaré Probab. Statist. 26 219-260.
29. NAGAEV, A. V. (1969). Limit theorems for large deviations when Cramér's conditions are violated. Izv. Akad. Nauk UzSSR Ser. Fiz-Mat. Nauk 6 17-22. (In Russian.)
30. NAGAEV, S. V. (1979). Large deviations of sums independent random variables. Ann. Probab. 7 745-789.
31. PETROV, V. V. (1995). Limit Theorems of Probability Theory. Oxford Univ. Press.
32. RESNICK, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.
33. RESNICK, S. I. and WILLEKENS, E. (1991). Moving averages with random coefficients and random coefficient autoregressive models. Commun. Statistics: Stochastic Models 7 511-525.
34. ROSENBLATT, M. (1956). A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42 43-47.
35. SAMORODNITSKY, G. (2002). Long Range Dependence, Heavy Tails and Rare Events. MaPhySto Lecture Notes. Available at http://www.maphysto.dk/.
36. SAMORODNITSKY, G. and TAQQU, M. S. (1994). Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance. Chapman and Hall, London.
37. STRAUMANN, D. (2003). Estimation in conditonally heteroscedastic time series models. Ph.D. thesis, Institute of Mathematical Science, Univ. Copenhagen.
38. STRAUMANN, D. and MIKOSCH, T. (2005). Quasi-maximum likelihood estimation in heteroscedastic time series: A stochastic recurrence equations approach. Ann. Statist. To appear.