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Convolutions of heavy-tailed random variables and applications to portfolio diversification and MA(1) time series

Part of: Distribution theory - Probability Limit theorems

Published online by Cambridge University Press:  01 July 2016

Jaap L. Geluk ,
Liang Peng  and
Casper G. de Vries
Jaap L. Geluk*
Affiliation:
Erasmus University Rotterdam
Liang Peng*
Affiliation:
The Australian National University
Casper G. de Vries*
Affiliation:
Erasmus University Rotterdam
*
∗ Postal address: Econometric Institute, Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands. Email address: [email protected]
∗∗ Postal address: Centre for Mathematics and its Applications, The Australian National University, Canberra, ACT 0200, Australia.
∗∗∗ Postal address: Department of Economics, Erasmus University Rotterdam, 3000 DR Rotterdam, The Netherlands.
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Abstract

Suppose X1,X2 are independent random variables satisfying a second-order regular variation condition on the tail-sum and a balance condition on the tails. In this paper we give a description of the asymptotic behaviour as t → ∞ for P(X1 + X2 > t). The result is applied to the problem of risk diversification in portfolio analysis and to the estimation of the parameter in a MA(1) model.

Keywords

Heavytailsregular variationportfolio diversificationconvolutiontime series

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60E05: Distributions
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 4 , December 2000 , pp. 1011 - 1026
Copyright
Copyright © Applied Probability Trust 2000 

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References

Bingham, N., Goldie, C. and Teugels, J. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Copeland, T. E. and Weston, J. F. (1983). Financial Theory and Corporate Policy. Addison-Wesley, Reading.Google Scholar
Dacorogna, M. M., Muller, U. A., Pictet, O. V. and de Vries, C. G. (1998). Extremal Forex returns in extremely large data sets. Submitted.Google Scholar
Danielsson, J. and de Vries, C. G. (1997). Tail index and quantile estimation with very high frequency data. J. Empirical Finance 4, 241–257.CrossRefGoogle Scholar
Danielsson, J. and de Vries, C. G. (2000). Value-at-risk and extreme returns. To appear in Ann. Économ. Statist. 60.Google Scholar
Datta, S. and McCormick, W. P. (1998). Inference for the tail parameters of a linear process with heavy tail innovations. Ann. Inst. Statist. Math. 50, 337–359.Google Scholar
Davis, R. A., Chen, M. and Dunsmuir, W. T. M. (1995). Inference for MA(1) processes with a root on or near the unit circle. Prob. Math. Statist. 15, 227–242.Google Scholar
Davis, R. A. and Dunsmuir, W. T. M. (1996). Maximum likelihood estimation for MA(1) processes with a root on or near the unit circle. Econometric Theory 12, 1–29.CrossRefGoogle Scholar
Davis, R. A. and Mikosch, T. (1998). Gaussian likelihood-based inference for non-invertible MA(1) processes with mathrm Salpha Snoise. Stoch. Proc. Appl. 77, 99–122.CrossRefGoogle Scholar
de Haan, L. and Pereira, T. T. (1999). Estimating the index of a stable distribution. Statist. Prob. Lett. 41, 39–55.Google Scholar
Dowd, K. (1998). Beyond Value at Risk, the New Science of Risk Management. John Wiley, Chichester.Google Scholar
Drees, H. and Kaufmann, E. (1998). Selecting the optimal sample fraction in univariate extreme value estimation. Stoch. Proc. Appl. 75, 149–172.Google Scholar
Fama, E. F. and Miller, M. H. (1972). The Theory of Finance. Dryden Press, Hinsdale.Google Scholar
Feller, W. (1971). An Introduction to Probability and its Applications, Vol. II. John Wiley, New York.Google Scholar
Geluk, J., de Haan, L., Resnick, S. and Starica, C. (1997). Second order regular vatiation, convolution and the central limit theorem. Stoch. Proc. Appl. 9, 139–159.Google Scholar
Geluk, J. and Peng, L. (2000). An adaptive optimal estimate of the tail index for MA(1) time series. Statist. Prob. Lett. 46, 217–227.CrossRefGoogle Scholar
Hall, P. and Weissman, I. (1997). On the estimation of extreme tail probabilities. Ann. Statist. 25, 1311–1326.CrossRefGoogle Scholar
Hill, B. M. (1975). A simple general approach to inference about the tail of a distribution. Ann. Statist. 3, 1163–1174.Google Scholar
Jorion, P. (1997). Value at Risk. Irwin, Chicago.Google Scholar
Lii, K. and Rosenblatt, M. (1982). Deconvolution and estimation of transfer function phase and coefficients for nongaussian linear processes. Ann. Statist. 10, 1195–1208.Google Scholar
Lii, K. and Rosenblatt, M. (1992). An approximate maximum likelihood estimation for non-Gaussian non-minimum phase moving average processes. J. Multivar. Anal. 43, 272–299.Google Scholar
Longin, F. M. (2000). From value-at-risk to stress testing: the extreme value approach. J. Banking Finance 24, 1097–1130.CrossRefGoogle Scholar
Mason, D. M. (1982). Laws of large numbers for sums of extreme values. Ann. Prob. 10, 754–764.Google Scholar
Resnick, S. and Starica, C. (1997). Asymptotic behavior of Hill's estimator for autoregressive data. Comm. Statist. Stoch. Models 13, 703–721.Google Scholar