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 Probability Surveys > Vol. 11 (2014) open journal systems 


Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps

Filip Lindskog, KTH Royal Institute of Technology, Stockholm
Sidney I. Resnick, Cornell University, School of Operations Research and Information Engineering
Joyjit Roy, Cornell University, School of Operations Research and Information Engineering


Abstract
We develop a framework for regularly varying measures on complete separable metric spaces \(\mathbb{S}\) with a closed cone \(\mathbb{C}\) removed, extending material in [15, 24]. Our framework provides a flexible way to consider hidden regular variation and allows simultaneous regular-variation properties to exist at different scales and provides potential for more accurate estimation of probabilities of risk regions. We apply our framework to iid random variables in \(\mathbb{R}_+^\infty\)\(\) with marginal distributions having regularly varying tails and to càdlàg Lèvy processes whose Lèvy measures have regularly varying tails. In both cases, an infinite number of regular-variation properties coexist distinguished by different scaling functions and state spaces.

AMS 2000 subject classifications: 28A33,60G17,60G51,60G70

Keywords: Regular variation, multivariate heavy tails, hidden regular variation, tail estimation, L'evy process

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Lindskog, Filip, Resnick, Sidney I., Roy, Joyjit, Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps, Probability Surveys, 11, (2014), 270-314 (electronic). DOI: 10.1214/14-PS231.

References

[1]    Applebaum, D., Lévy Processes and Stochastic Calculus, volume 116 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 2009. ISBN 978-0-521-73865-1. URL http://dx.doi.org/10.1017/CBO9780511809781. MR2512800

[2]    Balkema, A. A., Monotone Transformations and Limit Laws. Mathematisch Centrum, Amsterdam, 1973. Mathematical Centre Tracts, No. 45. MR0334307

[3]    Balkema, A. A. and Embrechts, P., High Risk Scenarios and Extremes: A Geometric Approach. European Mathematical Society, 2007. MR2372552

[4]    Barczy, M. and Pap, G., Portmanteau theorem for unbounded measures. Statistics & Probability Letters, 76(17):1831–1835, 2006. MR2271177

[5]    Bertoin, J., Lévy Processes, volume 121 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1996. ISBN 0-521- 56243-0. MR1406564

[6]    Billingsley, P., Convergence of Probability Measures. John Wiley & Sons Inc., New York, second edition, 1999. ISBN 0-471-19745-9. A Wiley-Interscience Publication. MR1700749

[7]    Billingsley, P., Probability and Measure. Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., Hoboken, NJ, 2012. ISBN 978-1-118-12237-2. Anniversary edition [of MR1324786], With a foreword by Steve Lalley and a brief biography of Billingsley by Steve Koppes. MR2893652

[8]    Bingham, N. H., Goldie, C. M., and Teugels, J. L., Regular Variation, volume 27 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1989. ISBN 0-521-37943-1. MR1015093

[9]    Breiman, L., Probability, volume 7 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. ISBN 0-89871-296-3. URL http://dx.doi.org/10.1137/1.9781611971286. Corrected reprint of the 1968 original. MR1163370

[10]    Bruun, J. T. and Tawn, J. A., Comparison of approaches for estimating the probability of coastal flooding. J. R. Stat. Soc., Ser. C, Appl. Stat., 47(3):405–423, 1998.

[11]    Coles, S. G., Heffernan, J. E., and Tawn, J. A., Dependence measures for extreme value analyses. Extremes, 2(4):339–365, 1999.

[12]    Daley, D. J. and Vere-Jones, D., An Introduction to the Theory of Point Processes. Vol. I. Probability and Its Applications (New York). Springer-Verlag, New York, second edition, 2003. ISBN 0-387-95541-0. Elementary theory and methods. MR1950431

[13]    Das, B. and Resnick, S. I., Conditioning on an extreme component: Model consistency with regular variation on cones. Bernoulli, 17(1):226–252, 2011. ISSN 1350-7265. MR2797990

[14]    Das, B. and Resnick, S. I., Detecting a conditional extreme value model. Extremes, 14(1):29–61, 2011. MR2775870

[15]    Das, B., Mitra, A., and Resnick, S. I., Living on the multi-dimensional edge: Seeking hidden risks using regular variation. Advances in Applied Probability, 45(1):139–163, 2013. ArXiv e-prints 1108.5560. MR3077544

[16]    de Haan, L., On Regular Variation and Its Application to the Weak Convergence of Sample Extremes. Mathematisch Centrum Amsterdam, 1970. MR0286156

[17]    de Haan, L. and Ferreira, A., Extreme Value Theory: An Introduction. Springer-Verlag, New York, 2006. MR2234156

[18]    Dudley, R. M., Real Analysis and Probability, volume 74 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2002. ISBN 0-521-00754-2. URL http://dx.doi.org/10.1017/CBO9780511755347. Revised reprint of the 1989 original. MR1932358

[19]    Embrechts, P., Klüppelberg, C., and Mikosch, T., Modelling Extremal Events for Insurance and Finance. Springer-Verlag, Berlin, 2003. 4th corrected printing. MR1458613

[20]    Geluk, J. L. and de Haan, L., Regular Variation, Extensions and Tauberian Theorems, volume 40 of CWI Tract. Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1987. ISBN 90-6196-324-9.

[21]    Heffernan, J. E. and Resnick, S. I., Hidden regular variation and the rank transform. Adv. Appl. Prob., 37(2):393–414, 2005. MR2144559

[22]    Heffernan, J. E. and Tawn, J. A., A conditional approach for multivariate extreme values (with discussion). JRSS B, 66(3):497–546, 2004. MR2088289

[23]    Hult, H. and Lindskog, F., Extremal behavior of regularly varying stochastic processes. Stochastic Process. Appl., 115(2):249–274, 2005. ISSN 0304-4149. MR2111194

[24]    Hult, H. and Lindskog, F., Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd) (N.S.), 80(94):121–140, 2006. ISSN 0350-1302. URL http://dx.doi.org/10.2298/PIM0694121H. MR2281910

[25]    Hult, H. and Lindskog, F., Extremal behavior of stochastic integrals driven by regularly varying Lévy processes. Ann. Probab., 35(1):309–339, 2007. ISSN 0091-1798. URL http://dx.doi.org/10.1214/009117906000000548. MR2303951

[26]    Hult, H., Lindskog, F., Mikosch, T., and Samorodnitsky, G., Functional large deviations for multivariate regularly varying random walks. Ann. Appl. Probab., 15(4):2651–2680, 2005. ISSN 1050-5164. MR2187307

[27]    Kyprianou, A. E., Fluctuations of Lévy Processes with Applications. Universitext. Springer, Heidelberg, second edition, 2014. ISBN 978-3-642-37631-3; 978-3-642-37632-0. URL http://dx.doi.org/10.1007/978-3-642-37632-0. Introductory lectures. MR3155252

[28]    Ledford, A. W. and Tawn, J. A., Statistics for near independence in multivariate extreme values. Biometrika, 83(1):169–187, 1996. ISSN 0006-3444. MR1399163

[29]    Ledford, A. W. and Tawn, J. A., Modelling dependence within joint tail regions. J. Roy. Statist. Soc. Ser. B, 59(2):475–499, 1997. ISSN 0035-9246. MR1440592

[30]    Maulik, K. and Resnick, S. I., Characterizations and examples of hidden regular variation. Extremes, 7(1):31–67, 2005. MR2201191

[31]    Maulik, K., Resnick, S. I., and Rootzén, H., Asymptotic independence and a network traffic model. J. Appl. Probab., 39(4):671–699, 2002. ISSN 0021-9002. MR1938164

[32]    Meerschaert, M. and Scheffler, H. P., Limit Distributions for Sums of Independent Random Vectors. John Wiley & Sons Inc., New York, 2001. ISBN 0-471-35629-8. MR1840531

[33]    Mitra, A. and Resnick, S. I., Hidden Regular Variation: Detection and Estimation. ArXiv e-prints, January 2010.

[34]    Mitra, A. and Resnick, S. I., Hidden regular variation and detection of hidden risks. Stochastic Models, 27(4):591–614, 2011. MR2854234

[35]    Mitra, A. and Resnick, S. I., Modeling multiple risks: Hidden domain of attraction. Extremes, 16:507–538, 2013. URL http:dx.doi.org/10.1007/s10687-013-0171-8. MR3133850

[36]    Peng, L., Estimation of the coefficient of tail dependence in bivariate extremes. Statist. Probab. Lett., 43(4):399–409, 1999. ISSN 0167-7152. MR1707950

[37]    Resnick, S. I., Point processes, regular variation and weak convergence. Adv. Applied Probability, 18:66–138, 1986. MR0827332

[38]    Resnick, S. I., A Probability Path. Birkhäuser, Boston, 1999. MR1664717

[39]    Resnick, S. I., Hidden regular variation, second order regular variation and asymptotic independence. Extremes, 5(4):303–336 (2003), 2002. ISSN 1386-1999. MR2002121

[40]    Resnick, S. I., Heavy Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering. Springer-Verlag, New York, 2007. ISBN 0-387-24272-4. MR2271424

[41]    Resnick, S. I., Multivariate regular variation on cones: Application to extreme values, hidden regular variation and conditioned limit laws. Stochastics: An International Journal of Probability and Stochastic Processes, 80(2):269–298, 2008. http://www.informaworld.com/10.1080/17442500701830423. MR2402168

[42]    Resnick, S. I., Extreme Values, Regular Variation and Point Processes. Springer, New York, 2008. ISBN 978-0-387-75952-4. Reprint of the 1987 original. MR2364939

[43]    Resnick, S. I. and Zeber, D., Transition kernels and the conditional extreme value model. Extremes, 17(2):263–287, 2014. ISSN 1386-1999. URL http://dx.doi.org/10.1007/s10687-014-0182-0. MR3207717

[44]    Royden, H. L., Real Analysis. Macmillan, third edition, 1988. MR1013117

[45]    Schlather, M., Examples for the coefficient of tail dependence and the domain of attraction of a bivariate extreme value distribution. Stat. Probab. Lett., 53(3):325–329, 2001. MR1841635

[46]    Seneta, E., Regularly Varying Functions. Springer-Verlag, New York, 1976. Lecture Notes in Mathematics, 508. MR0453936

[47]    Sibuya, M., Bivariate extreme statistics. Ann. Inst. Stat. Math., 11:195–210, 1960. MR0115241




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