## Bernoulli

• Bernoulli
• Volume 20, Number 2 (2014), 803-845.

### A Fourier analysis of extreme events

#### Abstract

The extremogram is an asymptotic correlogram for extreme events constructed from a regularly varying stationary sequence. In this paper, we define a frequency domain analog of the correlogram: a periodogram generated from a suitable sequence of indicator functions of rare events. We derive basic properties of the periodogram such as the asymptotic independence at the Fourier frequencies and use this property to show that weighted versions of the periodogram are consistent estimators of a spectral density derived from the extremogram.

#### Article information

Source
Bernoulli, Volume 20, Number 2 (2014), 803-845.

Dates
First available in Project Euclid: 28 February 2014

https://projecteuclid.org/euclid.bj/1393594007

Digital Object Identifier
doi:10.3150/13-BEJ507

Mathematical Reviews number (MathSciNet)
MR3178519

Zentralblatt MATH identifier
1321.60110

#### Citation

Mikosch, Thomas; Zhao, Yuwei. A Fourier analysis of extreme events. Bernoulli 20 (2014), no. 2, 803--845. doi:10.3150/13-BEJ507. https://projecteuclid.org/euclid.bj/1393594007

#### References

• [1] Andersen, T.G., Davis, R.A., Kreiss, J.P. and Mikosch, T. (2009). The Handbook of Financial Time Series. Springer: Heidelberg.
• [2] Bartkiewicz, K., Jakubowski, A., Mikosch, T. and Wintenberger, O. (2011). Stable limits for sums of dependent infinite variance random variables. Probab. Theory Related Fields 150 337–372.
• [3] Basrak, B., Davis, R.A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stochastic Process. Appl. 99 95–115.
• [4] 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.
• [5] Basrak, B. and Segers, J. (2009). Regularly varying multivariate time series. Stochastic Process. Appl. 119 1055–1080.
• [6] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge: Cambridge Univ. Press.
• [7] Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory Probab. Appl. 10 323–331.
• [8] Brillinger, D.R. (1981). Time Series: Data Analysis and Theory, 2nd ed. Oakland, CA: Holden-Day Inc.
• [9] Brockwell, P.J. and Davis, R.A. (1991). Time Series: Theory and Methods, 2nd ed. Springer Series in Statistics. New York: Springer.
• [10] Brockwell, P.J. and Davis, R.A. (2002). Introduction to Time Series and Forecasting, 2nd ed. Springer Texts in Statistics. New York: Springer.
• [11] 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.
• [12] Davis, R.A. and Mikosch, T. (2001). Point process convergence of stochastic volatility processes with application to sample autocorrelation. J. Appl. Probab. 38A 93–104.
• [13] Davis, R.A. and Mikosch, T. (2009). Extreme value theory for GARCH processes. In Handbook of Financial Time Series (T.G. Andersen, R.A. Davis, J.P. Kreiss and T. Mikosch, eds.) 187–200. Berlin: Springer.
• [14] Davis, R.A. and Mikosch, T. (2009). Extremes of stochastic volatility models. In Handbook of Financial Time Series (T.G. Andersen, R.A. Davis, J.P. Kreiss and T. Mikosch, eds.) 355–364. Berlin: Springer.
• [15] Davis, R.A. and Mikosch, T. (2009). The extremogram: A correlogram for extreme events. Bernoulli 15 977–1009.
• [16] Davis, R.A., Mikosch, T. and Cribben, I. (2012). Towards estimating extremal serial dependence via the bootstrapped extremogram. J. Econometrics 170 142–152.
• [17] Dette, H., Hallin, M., Kleya, T. and Volgusheva, S. (2011). On copulas, quantiles, ranks and spectra. An $L_{1}$-approach to spectral analysis. Working paper.
• [18] Dombry, C. and Eyi-Minko, F. (2012). Strong mixing properties of max-infinitely divisible random fields. Stochastic Process. Appl. 122 3790–3811.
• [19] Doukhan, P. (1994). Mixing: Properties and Examples. Lecture Notes in Statistics 85. New York: Springer.
• [20] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Applications of Mathematics (New York) 33. Berlin: Springer.
• [21] Fasen, V., Klüppelberg, C. and Schlather, M. (2010). High-level dependence in time series models. Extremes 13 1–33.
• [22] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. New York: Wiley.
• [23] Gradshteyn, I.S. and Ryzhik, I.M. (1980). Table of Integrals, Series, and Products. New York: Academic Press. Corrected and enlarged edition edited by Alan Jeffrey, Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V. Geronimus] and M. Yu. Tseytlin [M. Yu. Tseĭtlin], Translated from the Russian.
• [24] Grenander, U. and Rosenblatt, M. (1984). Statistical Analysis of Stationary Time Series, 2nd ed. New York: Chelsea.
• [25] Hagemann, A. (2011). Robust spectral analysis. Working paper, UICU.
• [26] Hannan, E.J. (1960). Time Series Analysis. Methuen’s Monographs on Applied Probability and Statistics. London: Methuen.
• [27] Hill, J.B. (2009). On functional central limit theorems for dependent, heterogeneous arrays with applications to tail index and tail dependence estimation. J. Statist. Plann. Inference 139 2091–2110.
• [28] Hill, J.B. (2011). Extremal memory of stochastic volatility with an application to tail shape inference. J. Statist. Plann. Inference 141 663–676.
• [29] Ibragimov, I.A. and Linnik, Y.V. (1971). Independent and Stationary Sequences of Random Variables. Groningen: Wolters-Noordhoff. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov, Translation from the Russian edited by J. F. C. Kingman.
• [30] Jakubowski, A. (1993). Minimal conditions in $p$-stable limit theorems. Stochastic Process. Appl. 44 291–327.
• [31] Jakubowski, A. (1997). Minimal conditions in $p$-stable limit theorems. II. Stochastic Process. Appl. 68 1–20.
• [32] Jessen, A.H. and Mikosch, T. (2006). Regularly varying functions. Publ. Inst. Math. (Beograd) (N.S.) 80 171–192.
• [33] Kallenberg, O. (1983). Random Measures, 3rd ed. Berlin: Akademie-Verlag.
• [34] Leadbetter, M.R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer Series in Statistics. New York: Springer.
• [35] Ledford, A.W. and Tawn, J.A. (2003). Diagnostics for dependence within time series extremes. J. R. Stat. Soc. Ser. B Stat. Methodol. 65 521–543.
• [36] Lee, J. and Subba Rao, S. (2012). The quantile spectral density and comparison based tests for nonlinear time series. Working paper.
• [37] Mikosch, T. and Rezapur, M. (2013). Stochastic volatility models with possible extremal clustering. Bernoulli 19 1688–1713.
• [38] Mikosch, T. and Samorodnitsky, G. (2000). The supremum of a negative drift random walk with dependent heavy-tailed steps. Ann. Appl. Probab. 10 1025–1064.
• [39] Mikosch, T. and Stărică, C. (2000). Limit theory for the sample autocorrelations and extremes of a GARCH $(1,1)$ process. Ann. Statist. 28 1427–1451.
• [40] Petrov, V.V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Oxford Studies in Probability 4. New York: The Clarendon Press Oxford Univ. Press.
• [41] Pham, T.D. and Tran, L.T. (1985). Some mixing properties of time series models. Stochastic Process. Appl. 19 297–303.
• [42] Priestley, M.B. (1981). Spectral Analysis and Time Series. London, New York: Academic Press.
• [43] Resnick, S.I. (1986). Point processes, regular variation and weak convergence. Adv. in Appl. Probab. 18 66–138.
• [44] Resnick, S.I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. New York: Springer.
• [45] Resnick, S.I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering. New York: Springer.