Probability Surveys

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

Filip Lindskog, Sidney I. Resnick, and Joyjit Roy

Full-text: Open access

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 Lévy processes whose Lévy measures have regularly varying tails. In both cases, an infinite number of regular-variation properties coexist distinguished by different scaling functions and state spaces.

Article information

Source
Probab. Surveys, Volume 11 (2014), 270-314.

Dates
First available in Project Euclid: 21 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.ps/1413896892

Digital Object Identifier
doi:10.1214/14-PS231

Mathematical Reviews number (MathSciNet)
MR3271332

Zentralblatt MATH identifier
1317.60007

Subjects
Primary: 28A33: Spaces of measures, convergence of measures [See also 46E27, 60Bxx] 60G17: Sample path properties 60G51: Processes with independent increments; Lévy processes 60G70: Extreme value theory; extremal processes

Keywords
Regular variation multivariate heavy tails hidden regular variation tail estimation M-convergence Lévy process

Citation

Lindskog, Filip; Resnick, Sidney I.; Roy, Joyjit. Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps. Probab. Surveys 11 (2014), 270--314. doi:10.1214/14-PS231. https://projecteuclid.org/euclid.ps/1413896892


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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.
  • [2] Balkema, A. A., Monotone Transformations and Limit Laws. Mathematisch Centrum, Amsterdam, 1973. Mathematical Centre Tracts, No. 45.
  • [3] Balkema, A. A. and Embrechts, P., High Risk Scenarios and Extremes: A Geometric Approach. European Mathematical Society, 2007.
  • [4] Barczy, M. and Pap, G., Portmanteau theorem for unbounded measures. Statistics & Probability Letters, 76(17):1831–1835, 2006.
  • [5] Bertoin, J., Lévy Processes, volume 121 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1996. ISBN 0-521- 56243-0.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [13] Das, B. and Resnick, S. I., Conditioning on an extreme component: Model consistency with regular variation on cones. Bernoulli, 17(1):226–252, 2011a. ISSN 1350-7265.
  • [14] Das, B. and Resnick, S. I., Detecting a conditional extreme value model. Extremes, 14(1):29–61, 2011b.
  • [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.
  • [16] de Haan, L., On Regular Variation and Its Application to the Weak Convergence of Sample Extremes. Mathematisch Centrum Amsterdam, 1970.
  • [17] de Haan, L. and Ferreira, A., Extreme Value Theory: An Introduction. Springer-Verlag, New York, 2006.
  • [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.
  • [19] Embrechts, P., Klüppelberg, C., and Mikosch, T., Modelling Extremal Events for Insurance and Finance. Springer-Verlag, Berlin, 2003. 4th corrected printing.
  • [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.
  • [22] Heffernan, J. E. and Tawn, J. A., A conditional approach for multivariate extreme values (with discussion). JRSS B, 66(3):497–546, 2004.
  • [23] Hult, H. and Lindskog, F., Extremal behavior of regularly varying stochastic processes. Stochastic Process. Appl., 115(2):249–274, 2005. ISSN 0304-4149.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [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.
  • [30] Maulik, K. and Resnick, S. I., Characterizations and examples of hidden regular variation. Extremes, 7(1):31–67, 2005.
  • [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.
  • [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.
  • [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.
  • [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.
  • [36] Peng, L., Estimation of the coefficient of tail dependence in bivariate extremes. Statist. Probab. Lett., 43(4):399–409, 1999. ISSN 0167-7152.
  • [37] Resnick, S. I., Point processes, regular variation and weak convergence. Adv. Applied Probability, 18:66–138, 1986.
  • [38] Resnick, S. I., A Probability Path. Birkhäuser, Boston, 1999.
  • [39] Resnick, S. I., Hidden regular variation, second order regular variation and asymptotic independence. Extremes, 5(4):303–336 (2003), 2002. ISSN 1386-1999.
  • [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.
  • [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, 2008a. http://www.informaworld.com/10.1080/17442500701830423.
  • [42] Resnick, S. I., Extreme Values, Regular Variation and Point Processes. Springer, New York, 2008b. ISBN 978-0-387-75952-4. Reprint of the 1987 original.
  • [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.
  • [44] Royden, H. L., Real Analysis. Macmillan, third edition, 1988.
  • [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.
  • [46] Seneta, E., Regularly Varying Functions. Springer-Verlag, New York, 1976. Lecture Notes in Mathematics, 508.
  • [47] Sibuya, M., Bivariate extreme statistics. Ann. Inst. Stat. Math., 11:195–210, 1960.