On convolution equivalence with applications

Bernoulli

Volumn 12, Issue 3, 2006, Pages 535-549

  Tang, Qihe ^a,b  

^a UNIVERSITY OF IOWA   (United States)

^b Gina Cody School of Engineering and Computer Science   (Canada)

Author keywords

Asymptotics; Class S( ); Endpoint; Poisson shot noise; Rapid variation; Stochastic equation; Uniformity

Indexed keywords

EID: 33750990844     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.3150/bj/1151525135     Document Type: Article

References (22)

1. Brémaud, P. (2000) An insensitivity property of Lundberg's estimate for delayed claims. J. Appl. Probab., 37(3), 914-917.
2. Chistyakov, V.P. (1964) A theorem on sums of independent positive random variables and its applications to branching random processes. Theory Probab. Appl., 9, 640-648.
3. Chover, J., Ney, P. and Wainger, S. (1973a) Functions of probability measures. J. Anal. Math., 26, 255-302.
4. Chover, J., Ney, P. and Wainger, S. (1973b) Degeneracy properties of subcritical branching processes. Ann. Probab., 1, 663-673.
5. Cline, D.B.H. (1987) Convolutions of distributions with exponential and subexponential tails. J. Austral. Math. Soc. Ser. A, 43(3), 347-365.
6. Cline, D.B.H. and Samorodnitsky, G. (1994) Subexponentiality of the product of independent random variables. Stochastic Process. Appl., 49(1), 75-98.
7. Embrechts, P. (1983) A property of the generalized inverse Gaussian distribution with some applications. J. Appl. Probab., 20(3), 537-544.
8. Embrechts, P. and Goldie, C.M. (1994) Perpetuities and random equations. In P. Mandl and M. Hušková (eds) Asymptotic Statistics: Proceedings of the Fifth Prague Symposium, pp. 75-86. Heidelberg: Physica.
9. Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance. Berlin: Springer-Verlag.
10. Goldie, C.M. (1991) Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab., 1(1), 126-166.
11. Klüppelberg, C. and Mikosch, T. (1995) Explosive Poisson shot noise processes with applications to risk reserves. Bernoulli, 1(1-2), 125-147.
12. Klüppelberg, C., Mikosch, T. and Schärf, A. (2003) Regular variation in the mean and stable limits for Poisson shot noise. Bernoulli, 9(3), 467-496.
13. Lund, R., McCormick, W.P. and Xiao, Y. (2004) Limiting properties of Poisson shot noise processes. J. Appl. Probab., 41(3), 911-918.
14. McCormick, W.P. (1997) Extremes for shot noise processes with heavy tailed amplitudes. J. Appl. Probab., 34(3), 643-656.
15. Pakes, A.G. (2004) Convolution equivalence and infinite divisibility. J. Appl. Probab., 41(2), 407-424.
16. Rogozin, B.A. (2000) On the constant in the definition of subexponential distributions. Theory Probab. Appl., 44(2), 409-412.
17. Rogozin, B.A. and Sgibnev, M.S. (1999) Banach algebras of measures on the line with given asymptotics of distributions at infinity. Siberian Math. J., 40(3), 565-576.
18. Ross, S.M. (1983) Stochastic Processes. New York: Wiley.
19. Samorodnitsky, G. (1998) Tail behaviour of some shot noise processes. In R.J. Adler, R.E. Feldman and M.S. Taqqu (eds), A Practical Guide to Heavy Tails: Statistical Techniques and Applications, pp. 473-486. Boston: Birkhäuser.
20. Shimura, T. and Watanabe, T. (2005) Infinite divisibility and generalized subexponentiality. Bernoulli, 11(3), 445-469.
21. Tang, Q. and Tsitsiashvili, G. (2003) Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stochastic Process. Appl., 108(2), 299-325.
22. Tang, Q. and Tsitsiashvili, G. (2004) Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments. Adv. in Appl. Probab., 36(4), 1278-1299.