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Polar decomposition of regularly varying time series in star-shaped metric spaces

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Abstract

There exist two ways of defining regular variation of a time series in a star-shaped metric space: either by the distributions of finite stretches of the series or by viewing the whole series as a single random element in a sequence space. The two definitions are shown to be equivalent. The introduction of a norm-like function, called modulus, yields a polar decomposition similar to the one in Euclidean spaces. The angular component of the time series, called angular or spectral tail process, captures all aspects of extremal dependence. The stationarity of the underlying series induces a transformation formula of the spectral tail process under time shifts.

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References

  • Basrak, B., Segers, J.: Regularly varying multivariate time series. Stochastic Process Appl 119(4), 1055–1080 (2009). doi:10.1016/j.spa.2008.05.004

    Article  MathSciNet  MATH  Google Scholar 

  • Basrak, B., Krizmanić, D., Segers, J.: A functional limit theorem for dependent sequences with infinite variance stable limits. Ann Probab 40(5), 2008–2033 (2012). doi:10.1214/11-AOP669

    Article  MathSciNet  MATH  Google Scholar 

  • Billingsley, P.: Probability and measure, 3rd edn. Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons Inc., New York, a Wiley-Interscience Publication (1995)

  • Billingsley, P.: Convergence of probability measures, 2nd edn. Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons Inc (1999)

  • Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27. Cambridge University Press, Cambridge (1987). doi:10.1017/CBO9780511721434

  • Davis, R.A., Mikosch, T.: The extremogram: a correlogram for extreme events. Bernoulli 15(4), 977–1009 (2009). doi:10.3150/09-BEJ213

    Article  MathSciNet  MATH  Google Scholar 

  • Davis, R.A., Mikosch, T., Zhao, Y.: Measures of serial extremal dependence and their estimation. Stochastic Process Appl 123(7), 2575–2602 (2013). doi:10.1016/j.spa.2013.03.014

    Article  MathSciNet  MATH  Google Scholar 

  • Dombry, C., Ribatet, M.: Functional regular variations, Pareto processes and peaks over threshold. Stat. Interface 8(1), 9–17 (2015). doi:10.4310/SII.2015.v8.n1.a2

  • Drees, H., Segers, J., Warchoł, M.: Statistics for tail processes of Markov chains. Extremes 18(3), 369–402 (2015). doi:10.1007/s10687-015-0217-1

    Article  MathSciNet  MATH  Google Scholar 

  • Giné, E., Hahn, M.G., Vatan, P.: Max-infinitely divisible and max-stable sample continuous processes. Probab. Theory Related Fields 87(2), 139–165 (1990). doi:10.1007/BF01198427

    Article  MathSciNet  MATH  Google Scholar 

  • de Haan, L., Lin, T.: On convergence toward an extreme value distribution in C[0,1]. Ann. Probab. 29(1), 467–483 (2001)

  • Hult, H., Lindskog, F.: Extremal behavior of regularly varying stochastic processes. Stochastic Process. Appl. 115(2), 249–274 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Hult, H., Lindskog, F.: Regular variation for measures on metric spaces. Publ. Inst. Math. 80(94), 121–140 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Janssen, A., Drees, H.: A stochastic volatility model with flexible extremal dependence structure. Bernoulli 22(3), 1448–1490 (2016). doi:10.3150/15-BEJ699

  • Janssen, A., Segers, J.: Markov tail chains. J. Appl. Probab. 51(4), 1133–1153 (2014). doi:10.1239/jap/1421763332

    Article  MathSciNet  MATH  Google Scholar 

  • Kuelbs, J., Mandrekar, V.: Domains of attraction of stable measures on a Hilbert space. Studia. Math. 50, 149–162 (1974)

    MathSciNet  MATH  Google Scholar 

  • Kulik, R., Soulier, P.: Heavy tailed time series with extremal independence. Extremes 18(2), 273–299 (2015). doi:10.1007/s10687-014-0213-x

    Article  MathSciNet  MATH  Google Scholar 

  • Leadbetter, M.R.: Extremes and local dependence in stationary sequences. Z. Wahrsch. Verw. Gebiete. 65(2), 291–306 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  • Lindskog, F., Resnick, S., Roy, J.: Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps. Probab. Surv. 11, 270–314 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Mandrekar, V., Zinn, J.: Central limit problem for symmetric case: convergence to non-Gaussian laws. Studia Math. 67(3), 279–296 (1980)

    MathSciNet  MATH  Google Scholar 

  • Meinguet, T.: Maxima of moving maxima of continuous functions. Extremes 15(3), 267–297 (2012). doi:10.1007/s10687-011-0136-8

    Article  MathSciNet  MATH  Google Scholar 

  • Meinguet, T., Segers, J.: Regularly varying time series in Banach spaces. arXiv:1001.3262 [mathPR] (2010)

  • Mikosch, T., Wintenberger, O.: The cluster index of regularly varying sequences with applications to limit theory for functions of multivariate Markov chains. Probab. Theory Relat. Fields 159(1-2), 157–196 (2014). doi:10.1007/s00440-013-0504-1

    Article  MathSciNet  MATH  Google Scholar 

  • Molchanov, I.: Theory of random sets. Probability and its Applications (New York), Springer-Verlag London, Ltd., London (2005)

  • Pollard, D.: A user’s guide to measure theoretic probability, Cambridge Series in Statistical and Probabilistic Mathematics, vol 8. Cambridge University Press (2002)

  • Resnick, S.: Hidden regular variation, second order regular variation and asymptotic independence. Extremes 5(4), 303–336 (2003). doi:10.1023/A:1025148622954

    Article  MathSciNet  MATH  Google Scholar 

  • Resnick, S.I.: Point processes, regular variation and weak convergence. Adv. Appl. Probab. 18(1), 66–138 (1986). doi:10.2307/1427239

    Article  MathSciNet  MATH  Google Scholar 

  • Resnick, S.I.: Extreme values, regular variation, and point processes, Applied Probability. A Series of the Applied Probability Trust, vol. 4. Springer-Verlag, New York (1987). doi:10.1007/978-0-387-75953-1

  • Resnick, S.I.: Heavy-tail phenomena. Springer Series in Operations Research and Financial Engineering, Springer, New York, probabilistic and statistical modeling (2007)

  • Samorodnitsky, G., Owada, T.: Tail measures of stochastic processes or random fields with regularly varying tails. Tech. rep., Cornell University, Ithaca, NY.. https://7c4299fd-a-62cb3a1a-s-sites.googlegroups.com/site/takashiowada54/files/tail.measure0103.pdf (2012)

  • van der Vaart, A.W., Wellner, J.A.: Weak convergence and empirical processes. Springer Series in Statistics, Springer-Verlag (1996)

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Authors and Affiliations

  1. Université catholique de Louvain, ISBA, Voie du Roman Pays 20, 1348, Louvain-la-Neuve, Belgium

    Johan Segers & Yuwei Zhao

  2. ING Bank N.V., Postbus 1800 1000 BV, Amsterdam, Netherlands

    Thomas Meinguet

Authors
  1. Johan Segers
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  2. Yuwei Zhao
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  3. Thomas Meinguet
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Corresponding author

Correspondence to Yuwei Zhao.

Additional information

The first author’s research is supported by contract nr. 07/12-045 of the Projet d’Actions de Recherche Concerté es of the Communauté française de Belgique, granted by the Académie universitaire Louvain and the second author’s research is supported by IAP research network grant nr. P7/06 of the Belgian government (Belgian Science Policy).

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Segers, J., Zhao, Y. & Meinguet, T. Polar decomposition of regularly varying time series in star-shaped metric spaces. Extremes 20, 539–566 (2017). https://doi.org/10.1007/s10687-017-0287-3

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