zbMATH — the first resource for mathematics

Tail measure and spectral tail process of regularly varying time series. (English) Zbl 1404.60074
Summary: The goal of this paper is an exhaustive investigation of the link between the tail measure of a regularly varying time series and its spectral tail process, independently introduced in [T. Owada and G. Samorodnitsky, “Tail measures of stochastic processes or random fields with regularly varying tails”, Preprint] and [B. Basrak and J. Segers, Stochastic Processes Appl. 119, No. 4, 1055–1080 (2009; Zbl 1161.60319)]. Our main result is to prove in an abstract framework that there is a one-to-one correspondence between these two objects, and given one of them to show that it is always possible to build a time series of which it will be the tail measure or the spectral tail process. For nonnegative time series, we recover results explicitly or implicitly known in the theory of max-stable processes.

60G70 Extreme value theory; extremal stochastic processes
Full Text: DOI Euclid
[1] Aaronson, J. (1997). An Introduction to Infinite Ergodic Theory. Mathematical Surveys and Monographs50. Amer. Math. Soc., Providence, RI.
[2] Basrak, B., Krizmanić, D. and Segers, J. (2012). A functional limit theorem for dependent sequences with infinite variance stable limits. Ann. Probab.40 2008–2033. · Zbl 1295.60041
[3] Basrak, B. and Planinić, H. (2018). A note on vague convergence of measures. Available at arXiv:1803.07024.
[4] Basrak, B. and Segers, J. (2009). Regularly varying multivariate time series. Stochastic Process. Appl.119 1055–1080. · Zbl 1161.60319
[5] Basrak, B. and Tafro, A. (2016). A complete convergence theorem for stationary regularly varying multivariate time series. Extremes19 549–560. · Zbl 1357.60034
[6] Buhl, S. and Klüppelberg, C. (2016). Anisotropic Brown–Resnick space–time processes: Estimation and model assessment. Extremes19 627–660. · Zbl 1357.62279
[7] Chernick, M. R., Hsing, T. and McCormick, W. P. (1991). Calculating the extremal index for a class of stationary sequences. Adv. in Appl. Probab.23 835–850. · Zbl 0741.60042
[8] Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes. Vol. I: Elementary Theory and Methods, 2nd ed. Springer, New York. · Zbl 1026.60061
[9] Davis, R. A. and Hsing, T. (1995). Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Probab.23 879–917. · Zbl 0837.60017
[10] Davis, R. A., Klüppelberg, C. and Steinkohl, C. (2013). Statistical inference for max-stable processes in space and time. J. R. Stat. Soc. Ser. B. Stat. Methodol.75 791–819.
[11] de Haan, L. (1984). A spectral representation for max-stable processes. Ann. Probab.12 1194–1204. · Zbl 0597.60050
[12] Dieker, A. B. and Mikosch, T. (2015). Exact simulation of Brown–Resnick random fields at a finite number of locations. Extremes18 301–314. · Zbl 1319.60108
[13] Dombry, C. and Kabluchko, Z. (2017). Ergodic decompositions of stationary max-stable processes in terms of their spectral functions. Stochastic Process. Appl.127 1763–1784. · Zbl 1367.60056
[14] Dudley, R. M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics74. Cambridge Univ. Press, Cambridge. Revised reprint of the 1989 original. · Zbl 1023.60001
[15] Hashorva, E. (2018). Representations of max-stable processes via exponential tilting. Stochastic Process. Appl.128 2952–2978. · Zbl 1405.60071
[16] Hult, H. and Lindskog, F. (2006). Regular variation for measures on metric spaces. Publ. Inst. Math. (Beograd) (N.S.) 80 121–140. · Zbl 1164.28005
[17] Janßen, A. (2017). Spectral tail processes and max-stable approximations of multivariate regularly varying time series. Available at arXiv:1704.06179.
[18] Janssen, A., Mikosch, T., Rezapour, M. and Xie, X. (2018). The eigenvalues of the sample covariance matrix of a multivariate heavy-tailed stochastic volatility model. Bernoulli24 1351–1393. · Zbl 1414.62368
[19] Kallenberg, O. (2017). Random Measures, Theory and Applications. Probability Theory and Stochastic Modelling77. Springer, Cham. · Zbl 1376.60003
[20] Kulik, R. (2016). Editorial: Special issue on time series extremes. Extremes19 463–466. · Zbl 1355.00024
[21] Lindskog, F., Resnick, S. I. and Roy, J. (2014). Regularly varying measures on metric spaces: Hidden regular variation and hidden jumps. Probab. Surv.11 270–314. · Zbl 1317.60007
[22] Mikosch, T. and Wintenberger, O. (2016). A large deviations approach to limit theory for heavy-tailed time series. Probab. Theory Related Fields166 233–269. · Zbl 1350.60024
[23] Planinić, H. and Soulier, P. (2018). The tail process revisited. Extremes. DOI:10.1007/s10687-018-0312-1.
[24] Segers, J., Zhao, Y. and Meinguet, T. (2017). Polar decomposition of regularly varying time series in star-shaped metric spaces. Extremes20 539–566. · Zbl 1387.60087
[25] Smith, R.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.