Bernoulli

  • Bernoulli
  • Volume 16, Number 4 (2010), 995-1015.

Weak convergence of the function-indexed integrated periodogram for infinite variance processes

Sami Umut Can, Thomas Mikosch, and Gennady Samorodnitsky

Full-text: Open access

Abstract

In this paper, we study the weak convergence of the integrated periodogram indexed by classes of functions for linear processes with symmetric $α$-stable innovations. Under suitable summability conditions on the series of the Fourier coefficients of the index functions, we show that the weak limits constitute $α$-stable processes which have representations as infinite Fourier series with i.i.d. $α$-stable coefficients. The cases $α ∈ (0, 1)$ and $α ∈ [1, 2)$ are dealt with by rather different methods and under different assumptions on the classes of functions. For example, in contrast to the case $α ∈ (0, 1)$, entropy conditions are needed for $α ∈ [1, 2)$ to ensure the tightness of the sequence of integrated periodograms indexed by functions. The results of this paper are of additional interest since they provide limit results for infinite mean random quadratic forms with particular Toeplitz coefficient matrices.

Article information

Source
Bernoulli, Volume 16, Number 4 (2010), 995-1015.

Dates
First available in Project Euclid: 18 November 2010

Permanent link to this document
https://projecteuclid.org/euclid.bj/1290092893

Digital Object Identifier
doi:10.3150/10-BEJ253

Mathematical Reviews number (MathSciNet)
MR2759166

Zentralblatt MATH identifier
1207.62173

Keywords
asymptotic theory empirical spectral distribution entropy infinite variance process integrated periodogram linear process random quadratic form spectral analysis stable process time series weighted integrated periodogram

Citation

Can, Sami Umut; Mikosch, Thomas; Samorodnitsky, Gennady. Weak convergence of the function-indexed integrated periodogram for infinite variance processes. Bernoulli 16 (2010), no. 4, 995--1015. doi:10.3150/10-BEJ253. https://projecteuclid.org/euclid.bj/1290092893


Export citation

References

  • [1] Adler, R.J., Feldman, R.E. and Taqqu, M.S. (eds.) (1998). A Practical Guide to Heavy Tails. Boston: Birkhäuser.
  • [2] Bartlett, M.S. (1954). Problemes de l’analyse spectrale des séries temporelles stationnaires. Publ. Inst. Statist. Uni. Paris III-3 119–134.
  • [3] Billingsley, P. (1968). Convergence of Probability Measures. New York: Wiley.
  • [4] Brockwell, P. and Davis, R. (1991). Time Series: Theory and Methods, 2nd ed. New York: Springer.
  • [5] Can, S.U., Mikosch, T. and Samorodnitsky, G. (2009). Weak convergence of the function-indexed integrated periodogram for infinite variuance processes. Extended technical report. Available at www.math.ku.dk/~mikosch.
  • [6] Crovella, M. and Bestavros, A. (1996). Self-similarity in world wide web traffic: Evidence and possible causes. In Proceedings of the 1996 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems 24 160–169.
  • [7] Crovella, M., Bestavros, A. and Taqqu, M.S. (1996). Heavy-tailed probability distributions in the world wide web. Preprint.
  • [8] Dahlhaus, R. (1988). Empirical spectral processes and their applications to time series analysis. Stochastic Process. Appl. 30 69–83.
  • [9] Dahlhaus, R. and Polonik, W. (2002). Empirical processes and nonparametric maximum likelihood estimation for time series. In Empirical Process Techniques for Dependent Data (H.G. Dehling, T. Mikosch and M. Sørensen, eds.) 275–298. Boston: Birkhäuser.
  • [10] Davis, R.A. and Mikosch, T. (2009). Extreme value theory for GARCH processes. In The Handbook of Financial Time Series (T.G. Andersen, R.A. Davis, J.-P. Kreiss and T. Mikosch, eds.) 187–200. Heidelberg: Springer.
  • [11] Davis, R.A. and Mikosch, T. (2009). Extremes of stochastic volatility models. In The Handbook of Financial Time Series (T.G. Andersen, R.A. Davis, J.-P. Kreiss and T. Mikosch, eds.) 355–364. Heidelberg: Springer.
  • [12] Davis, R.A. and Resnick, S.I. (1986). Limit theory for the sample covariance and correlation functions of moving averages. Ann. Statist. 14 533–558.
  • [13] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Finance and Insurance. Berlin: Springer.
  • [14] Faÿ, G., Gonzalez-Arevalo, B., Mikosch, T. and Samorodnitsky, G. (2006). Modeling teletraffic arrivals by a Poisson cluster process. Qesta 54 121–140.
  • [15] Feller, W. (1971). An Introduction to Probability Theory and Its Applications 2, 2nd ed. New York: Wiley.
  • [16] Grenander, U. and Rosenblatt, M. (1984). Statistical Analysis of Stationary Time Series, 2nd ed. New York: Chelsea.
  • [17] Kagan, Y.Y. (1997). Earthquake size distribution and earthquake insurance. Comm. Statist. Stochastic Models 13 775–797.
  • [18] Kokoszka, P. and Mikosch, T. (1997). The integrated periodogram for long-memory processes with finite or infinite variance. Stochastic Process. Appl. 66 55–78.
  • [19] Kokoszka, P. and Taqqu, M.S. (1996). Parameter estimation for infinite variance fractional ARIMA. Ann. Statist. 24 1880–1913.
  • [20] Klüppelberg, C. and Mikosch, T. (1996). The integrated periodogram for stable processes. Ann. Statist. 24 1855–1879.
  • [21] Klüppelberg, C. and Mikosch, T. (1996). Self-normalized and randomly centred spectral estimates. In Proceedings of the Athens International Conference on Applied Probability and Time Series, vol. 2: Time Series (C.C. Heyde, Yu.V. Prokhorov, R. Pyke and S.T. Rachev, eds.) 259–271. Berlin: Springer.
  • [22] Kwapień, S. and Woyczyński, W.A. (1992). Random Series and Stochastic Integrals: Single and Multiple. Basel: Birkhäuser.
  • [23] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces: Isoperimetry and Processes. Berlin: Springer.
  • [24] Leland, W.E., Willinger, W., Taqqu, M.S. and Wilson, D.V. (1993). On the self-similar nature of Ethernet traffic. Computer Communications Review 23 183–193.
  • [25] Mikosch, T. (1998). Periodogram estimates from heavy-tailed data. In A Practical Guide to Heavy Tails: Statistical Techniques for Analysing Heavy-Tailed Distributions (R.A. Adler, R. Feldman and M.S. Taqqu, eds.) 241–258. Boston: Birkhäuser.
  • [26] Mikosch, T. (2003). Modelling dependence and tails of financial time series. In Extreme Values in Finance, Telecommunications and the Environment (B. Finkenstädt and H. Rootzén, eds.) 185–286. London: Chapman & Hall.
  • [27] Mikosch, T., Gadrich, T., Klüppelberg, C. and Adler, R.J. (1995). Parameter estimation for ARMA models with infinite variance innovations. Ann. Statist. 23 305–326.
  • [28] Mikosch, T. and Norvaiša, R. (1997). Uniform convergence of the empirical spectral distribution function. Stochastic Process. Appl. 70 85–114.
  • [29] Mikosch, T., Resnick, S. and Samorodnitsky, G. (2000). The maximum of the periodogram for a heavy-tailed sequence. Ann. Appl. Probab. 28 885–908.
  • [30] Petrov, V.V. (1995). Limit Theorems of Probability Theory. Oxford: Oxford Univ. Press.
  • [31] Pollard, D. (1984). Convergence of Stochastic Processes. Berlin: Springer.
  • [32] Priestley, M. (1981). Spectral Analysis and Time Series, I and II. New York: Academic Press.
  • [33] Resnick, S.I. (2006). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. New York: Springer.
  • [34] Rosiński, J. and Woyczyński, W.A. (1987). Multilinear forms in Pareto-like random variables and product random measures. Colloq. Math. 51 303–313.
  • [35] Samorodnitsky, G. and Taqqu, M. (1994). Stable Non-Gaussian Random Processes. New York: Chapman & Hall.
  • [36] van der Vaart, A.W. and Wellner, J.A. (1996). Weak Convergence and Empirical Processes. New York: Springer.
  • [37] Whittle, P. (1951). Hypothesis Testing in Time Series Analysis. Uppsala: Almqvist & Wicksel.
  • [38] Willinger, W., Taqqu, M.S., Sherman, R. and Wilson, D. (1995). Self-similarity through high variability: Statistical analysis of ethernet lan traffic at the source level. Proceedings of the ACM/SIGCOMM’95, Cambridge, MA. Computer Communications Review 25 100–113.
  • [39] Zygmund, A. (2002). Trigonometric Series, I and II, 3rd ed. Cambridge, UK: Cambridge Univ. Press.