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Maxima of Sums of Heavy-Tailed Random Variables

Published online by Cambridge University Press:  29 August 2014

K.W. Ng
Affiliation:
Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulatn Road, Hong Kong, E-mail:[email protected]
Q.H. Tang
Affiliation:
University of Amsterdam, Department of Quantitative Economics, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands, E-mail:[email protected]
H. Yang
Affiliation:
Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam Road, Hong Kong, E-mail:[email protected]
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Abstract

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In this paper, we investigate asymptotic properties of the tail probabilities of the maxima of partial sums of independent random variables. For some large classes of heavy-tailed distributions, we show that the tail probabilities of the maxima of the partial sums asymptotically equal to the sum of the tail probabilities of the individual random variables. Then we partially extend the result to the case of random sums. Applications to some commonly used risk processes are proposed. All heavy-tailed distributions involved in this paper are supposed on the whole real line.

Type
Articles
Copyright
Copyright © International Actuarial Association 2002

References

Berman, S.M. (1986) The supremum of a process with stationary independent and symmetric increments. Stochastic Process. Appl, 23(2): 281290.CrossRefGoogle Scholar
Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987) Regular Variation. Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Braverman, M. (1999) Remarks on suprema of Lévy processes with light tails. Statist. Probab. Lett., 43(1): 4148.CrossRefGoogle Scholar
Braverman, M. (2000) Suprema of compound Poisson processes with light tails. Stochastic Process. Appl., 90(1), 145156.CrossRefGoogle Scholar
Braverman, M. and Samorodnitsky, G. (1995) Functionals of infinitely divisible stochastic processes with exponential tails. Stochastic Process. Appl., 56(2): 207231.CrossRefGoogle Scholar
Chover, J., Ney, P. and Wainger, S. (1973) Functions of probability measures. J. Analyse Math., 26: 255302.CrossRefGoogle Scholar
Cline, D.B.H. (1986) Convolution tails, product tails and domains of attraction. Probab. Theory Relat. Fields., 72(4), 529557.CrossRefGoogle Scholar
Cline, D.B.H. (1987) Convolutions of distributions with exponential and subexponential tails. J. Austral. Math. Soc. Sen A, 43: 347365.CrossRefGoogle Scholar
Embrechts, P., Goldie, C.M. and Veraverbeke, N. (1979) Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 49: 335347.CrossRefGoogle Scholar
Embrechts, P. and Goldie, C.M. (1982) On convolution tails. Stock Proc. Appl, 13: 263278.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance. Springer, Berlin.CrossRefGoogle Scholar
Goldie, C.M. and Klüppelberg, C. (1998) Subexponential distributions. A practical Guide to Heavy-Tails: Statistical Techniques and Applications. Eds. Adler, R.J., Feldman, R.E., Taqqu, M.S.Birkhäuser.Google Scholar
Ng, K.W., Tang, Q.H., Yan, J.A. and Yang, H. (2001) Large deviation results for collective insurance risk processes with heavy-tailed claims and applications. Submitted.Google Scholar
Petrov, V.V. (1975) A generalization of an inequality of Lévy. Theory Prob. Appl, 20(1): 141145.CrossRefGoogle Scholar
Sgibnev, M.S. (1988) Banach algebras of measures of class S(γ). Siberian Math. J., 29: 647655.CrossRefGoogle Scholar
Sgibnev, M.S. (1996) On the distribution of the maxima of partial sums. Statistics & Probability Letters, 28: 235238.CrossRefGoogle Scholar
Smith, W.L. (1958) Renewal theory and its ramifications. J. Roy. Statist. Soc. Ser. B, 20: 243302.Google Scholar
Tang, Q.H., Su, C., Jiang, T. and Zhang, J.S. (2001) Large deviations for heavy-tailed random sums in compound renewal model. Stat. Prob. Letters, 52(1): 91100.CrossRefGoogle Scholar
Willekens, E. (1986) Subexponentiality on the real line. Technical Report, K.U. Leuven.Google Scholar
Willekens, E. (1987) On the supremum of an infinitely divisible process. Stochastic Process. Appl, 26(1): 173175.CrossRefGoogle Scholar
Yang, H. (1999) Non-exponential bounds for ruin probability with interest effect included. Scandinavian Actuarial Journal, 6679.CrossRefGoogle Scholar