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General inverse problems for regular variation

Part of: Limit theorems Distribution theory - Probability

Published online by Cambridge University Press:  30 March 2016

Ewa Damek ,
Thomas Mikosch ,
Jan Rosiński  and
Gennady Samorodnitsky
Ewa Damek
Affiliation:
University of Wrocław, Mathematical Institute, Wrocław University, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. Email address: ewa.damek@math.uni.worc.pl.
Thomas Mikosch
Affiliation:
Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark. Email address: mikosch@math.ku.dk.
Jan Rosiński
Affiliation:
Department of Mathematics, 227 Ayres Hall, University of Tennessee, Knoxville, TN 37996-1320, USA. Email address: rosinski@math.utk.edu.
Gennady Samorodnitsky
Affiliation:
School of Operations Research and Information Engineering, and Department of Statistics, Cornell University, 220 Rhodes Hall, Ithaca, NY 14853, USA. Email address: gs18@cornell.edu.
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Abstract

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Regular variation of distributional tails is known to be preserved by various linear transformations of some random structures. An inverse problem for regular variation aims at understanding whether the regular variation of a transformed random object is caused by regular variation of components of the original random structure. In this paper we build on previous work, and derive results in the multivariate case and in situations where regular variation is not restricted to one particular direction or quadrant.

Keywords

Regular variationinverse problemlinear processBreiman's resultrandom matrix

MSC classification

Primary: 60E05: Distributions
Secondary: 60F05: Central limit and other weak theorems

Information

Type
Part 6. Heavy tails
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014 , pp. 229 - 248
Copyright
Copyright © Applied Probability Trust 2014 

References

Basrak, B., Davis, R. A., and Mikosch, T. (2002a). A characterization of multivariate regular variation. Ann. Appl. Prob./ 12, 908–920.Google Scholar
Basrak, B., Davis, R. A., and Mikosch, T. (2002b). Regular variation of {GARCH} processes. Stoch. Process. Appl./ 99, 95–115.Google Scholar
Davis, R. A., and Mikosch, T. (2009a). Extreme value theory for GARCH processes. In Handbook of Financial Time Series, eds Andersen, T. G. et al., Springer, Berlin, pp. 355–364.Google Scholar
Davis, R. A., and Mikosch, T. (2009b). Extremes of stochastic volatility models. In Handbook of Financial Time Series, eds Andersen, T. G. et al., Springer, Berlin, pp. 355–364.CrossRefGoogle Scholar
Embrechts, P., Goldie, C. M., and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth./ 49, 335–347.Google Scholar
Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.Google Scholar
Hult, H., and Samorodnitsky, G. (2008). Tail probabilities for infinite series of regularly varying random vectors. Bernoulli 14, 838–864.Google Scholar
Hult, H., and Samorodnitsky, G. (2010). Large deviations for point processes based on stationary sequences with heavy tails. J. Appl. Prob./ 47, 1–40.CrossRefGoogle Scholar
Jacobsen, M., Mikosch, T., Rosiński, J., and Samorodnitsky, G. (2009). Inverse problems for regular variation of linear filters, a cancellation property for σ-finite measures and identification of stable laws. Ann. Appl. Prob./ 19, 210–242.Google Scholar
Kallenberg, O. (1983). Random Measures, 3rd edn. Akademie, Berlin.Google Scholar
Mikosch, T., and Samorodnitsky, G. (2000). The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Prob./ 10, 1025–1064.Google Scholar
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York.CrossRefGoogle Scholar
Resnick, S. I., and Willekens, E. (1991). Moving averages with random coefficients and random coefficient autoregressive models. Commun. Statist. Stoch. Models 7, 511–525.Google Scholar
Rudin, W. (1973). Functional Analysis. McGraw Hill, New York.Google Scholar