The Annals of Probability

A functional limit theorem for dependent sequences with infinite variance stable limits

Bojan Basrak, Danijel Krizmanić, and Johan Segers

Full-text: Open access

Abstract

Under an appropriate regular variation condition, the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non-Gaussian stable random variable. A functional version of this is known to be true as well, the limit process being a stable Lévy process. The main result in the paper is that for a stationary, regularly varying sequence for which clusters of high-threshold excesses can be broken down into asymptotically independent blocks, the properly centered partial sum process still converges to a stable Lévy process. Due to clustering, the Lévy triple of the limit process can be different from the one in the independent case. The convergence takes place in the space of càdlàg functions endowed with Skorohod’s $M_{1}$ topology, the more usual $J_{1}$ topology being inappropriate as the partial sum processes may exhibit rapid successions of jumps within temporal clusters of large values, collapsing in the limit to a single jump. The result rests on a new limit theorem for point processes which is of independent interest. The theory is applied to moving average processes, squared $\operatorname{GARCH}(1,1)$ processes and stochastic volatility models.

Article information

Source
Ann. Probab., Volume 40, Number 5 (2012), 2008-2033.

Dates
First available in Project Euclid: 8 October 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1349703314

Digital Object Identifier
doi:10.1214/11-AOP669

Mathematical Reviews number (MathSciNet)
MR3025708

Zentralblatt MATH identifier
1295.60041

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60G52: Stable processes
Secondary: 60G55: Point processes 60G70: Extreme value theory; extremal processes

Keywords
Convergence in distribution functional limit theorem GARCH mixing moving average partial sum point processes regular variation stable processes spectral processes stochastic volatility

Citation

Basrak, Bojan; Krizmanić, Danijel; Segers, Johan. A functional limit theorem for dependent sequences with infinite variance stable limits. Ann. Probab. 40 (2012), no. 5, 2008--2033. doi:10.1214/11-AOP669. https://projecteuclid.org/euclid.aop/1349703314


Export citation

References

  • [1] Aue, A., Berkes, I. and Horváth, L. (2008). Selection from a stable box. Bernoulli 14 125–139.
  • [2] Avram, F. and Taqqu, M. S. (1992). Weak convergence of sums of moving averages in the $\alpha$-stable domain of attraction. Ann. Probab. 20 483–503.
  • [3] 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.
  • [4] Basrak, B., Davis, R. A. and Mikosch, T. (2002). Regular variation of GARCH processes. Stochastic Process. Appl. 99 95–115.
  • [5] Basrak, B. and Segers, J. (2009). Regularly varying multivariate time series. Stochastic Process. Appl. 119 1055–1080.
  • [6] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Math. 121. Cambridge Univ. Press, Cambridge.
  • [7] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [8] Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
  • [9] Dabrowski, A. R. and Jakubowski, A. (1994). Stable limits for associated random variables. Ann. Probab. 22 1–16.
  • [10] Davis, R. and Resnick, S. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Probab. 13 179–195.
  • [11] Davis, R. A. (1983). Stable limits for partial sums of dependent random variables. Ann. Probab. 11 262–269.
  • [12] 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.
  • [13] Davis, R. A. and Mikosch, T. (1998). The sample autocorrelations of heavy-tailed processes with applications to ARCH. Ann. Statist. 26 2049–2080.
  • [14] Davis, R. A. and Mikosch, T. (2009). The extremogram: A correlogram for extreme events. Bernoulli 15 977–1009.
  • [15] Davis, R. A. and Mikosch, T. (2009). Probabilistic properties of stochastic volatility models. In Handbook of Financial Time Series (T. G. Anderson, R. A. Davis, J. P. Kreiss and T. Mikosch, eds.) 255–268. Springer.
  • [16] Denker, M. and Jakubowski, A. (1989). Stable limit distributions for strongly mixing sequences. Statist. Probab. Lett. 8 477–483.
  • [17] Durrett, R. and Resnick, S. I. (1978). Functional limit theorems for dependent variables. Ann. Probab. 6 829–846.
  • [18] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Applications of Mathematics (New York) 33. Springer, Berlin.
  • [19] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. Wiley, New York.
  • [20] Gnedenko, B. V. and Kolmogorov, A. N. (1954). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Cambridge, MA.
  • [21] Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 126–166.
  • [22] Gouëzel, S. (2004). Central limit theorem and stable laws for intermittent maps. Probab. Theory Related Fields 128 82–122.
  • [23] Herrndorf, N. (1985). A functional central limit theorem for strongly mixing sequences of random variables. Z. Wahrsch. Verw. Gebiete 69 541–550.
  • [24] Jakubowski, A. (1993). Minimal conditions in $p$-stable limit theorems. Stochastic Process. Appl. 44 291–327.
  • [25] Jakubowski, A. (1997). Minimal conditions in $p$-stable limit theorems. II. Stochastic Process. Appl. 68 1–20.
  • [26] Jakubowski, A. and Kobus, M. (1989). $\alpha$-stable limit theorems for sums of dependent random vectors. J. Multivariate Anal. 29 219–251.
  • [27] Kallenberg, O. (1983). Random Measures, 3rd ed. Akademie-Verlag, Berlin.
  • [28] Kallenberg, O. (1997). Foundations of Modern Probability. Springer, New York.
  • [29] Kolmogorov, A. N. and Rozanov, Y. A. (1960). On strong mixing conditions for stationary Gaussian process. Theory Probab. Appl. 5 204–208.
  • [30] Krizmanić, D. (2010). Functional limit theorems for weakly dependent regularly varying time series. Ph.D. thesis, Univ. Zagreb. Available at http://www.math.uniri.hr/~dkrizmanic/DKthesis.pdf.
  • [31] Leadbetter, M. R. and Rootzén, H. (1988). Extremal theory for stochastic processes. Ann. Probab. 16 431–478.
  • [32] LePage, R., Woodroofe, M. and Zinn, J. (1981). Convergence to a stable distribution via order statistics. Ann. Probab. 9 624–632.
  • [33] Meinguet, T. and Segers, J. (2010). Regularly varying time series in Banach space. Université catholique de Louvain, Institut de statistique DP1002. Available at http://arxiv.org/abs/1001.3262.
  • [34] Merlevède, F. and Peligrad, M. (2000). The functional central limit theorem under the strong mixing condition. Ann. Probab. 28 1336–1352.
  • [35] 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.
  • [36] Mori, T. (1977). Limit distributions of two-dimensional point processes generated by strong-mixing sequences. Yokohama Math. J. 25 155–168.
  • [37] Peligrad, M. and Utev, S. (2005). A new maximal inequality and invariance principle for stationary sequences. Ann. Probab. 33 798–815.
  • [38] Petrov, V. V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Oxford Studies in Probability 4. Clarendon, Oxford Univ. Press, New York.
  • [39] Pham, T. D. and Tran, L. T. (1985). Some mixing properties of time series models. Stochastic Process. Appl. 19 297–303.
  • [40] Resnick, S. I. (1986). Point processes, regular variation and weak convergence. Adv. in Appl. Probab. 18 66–138.
  • [41] Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.
  • [42] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Stud. Adv. Math. 68. Cambridge Univ. Press, Cambridge.
  • [43] Segers, J. (2003). Functionals of clusters of extremes. Adv. in Appl. Probab. 35 1028–1045.
  • [44] Segers, J. (2005). Approximate distributions of clusters of extremes. Statist. Probab. Lett. 74 330–336.
  • [45] Skorohod, A. V. (1957). Limit theorems for stochastic processes with independent increments. Theory Probab. Appl. 2 145–177.
  • [46] Sly, A. and Heyde, C. (2008). Nonstandard limit theorem for infinite variance functionals. Ann. Probab. 36 796–805.
  • [47] Smith, R. L. (1992). The extremal index for a Markov chain. J. Appl. Probab. 29 37–45.
  • [48] Tyran-Kamińska, M. (2010). Convergence to Lévy stable processes under some weak dependence conditions. Stochastic Process. Appl. 120 1629–1650.
  • [49] Tyran-Kamińska, M. (2010). Functional limit theorems for linear processes in the domain of attraction of stable laws. Statist. Probab. Lett. 80 975–981.
  • [50] Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer, New York.