Hostname: page-component-7dd5485656-g8tfn Total loading time: 0 Render date: 2025-10-31T18:32:30.621Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit Theorems for Aggregated Linear Processes

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  22 February 2016

M. Jirak
M. Jirak*
Affiliation:
Graz University of Technology
*
∗ Current address: Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we develop an asymptotic theory of aggregated linear processes, and determine in particular the limit distribution of a large class of linear and nonlinear functionals of such processes. Given a sample {Y 1 (N),…,Y n (N)} of the normalized N-fold aggregated process, we describe the limiting behavior of statistics T N,n = T N,n (Y 1 (N),…, Y n (N)) in both of the cases n/N(n) → 0 and N(n)/n → 0, assuming either a ‘limiting long- or short-memory’ condition on the underlying linear process.

Keywords

Aggregationshort memorylong memoryinvariance principlerandom coefficient MA(∞)martingale decomposition

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60G99: None of the above, but in this section

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 2 , June 2013 , pp. 520 - 544
Copyright
© Applied Probability Trust 

References

Anh, V. V., Heyde, C. C. and Leonenko, N. N. (2002). Dynamic models of long-memory processes driven by Lévy noise. J. Appl. Prob. 39, 730–747.CrossRefGoogle Scholar
Anh, V. V., Knopova, V. P. and Leonenko, N. N. (2004). Continuous-time stochastic processes with cyclical long-range dependence. Austral. N. Z. J. Statist. 46, 275–296.CrossRefGoogle Scholar
Anh, V. V., Leonenko, N. N. and McVinish, R. (2001). Models for fractional Riesz-Bessel motion and related processes. Fractals 9, 329–346.CrossRefGoogle Scholar
Arcones, M. A. (1994). Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann. Prob. 22, 2242–2274.CrossRefGoogle Scholar
Arcones, M. A. (2000). Distributional limit theorems over a stationary Gaussian sequence of random vectors. Stoch. Process. Appl. 88, 135–159.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. (2000). Superposition of Ornstein-Uhlenbeck type processes. Teor. Veroyat. Primen. 45, 289–311. English translation: Theory Prob. Appl. 45 (2001), 175-194.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. and Leonenko, N. N. (2005). Spectral properties of superpositions of Ornstein-Uhlenbeck type processes. Methodology Comput. Appl. Prob. 7, 335–352.CrossRefGoogle Scholar
Beran, J. and Ocker, D. (2000). Temporal aggregation of stationary and nonstationary farima (p, d, 0) models. CoFE discussion paper, University of Konstanz.Google Scholar
Beran, J., Ghosh, S. and Schützner, M. (2010). From short to long memory: aggregation and estimation. Comput. Statist. Data Anal. 54, 2432–2442.CrossRefGoogle Scholar
Celov, D., Leipus, R. and Philippe, A. (2007). Time series aggregation, disaggregation, and long memory. Liet. Mat. Rink. 47, 466–481. English translation: Lithuanian Math. J. 47, 379-393.Google Scholar
Celov, D., Leipus, R. and Philippe, A. (2010). Asymptotic normality of the mixture density estimator in a disaggregation scheme. J. Nonparametric Statist. 22, 425–442.CrossRefGoogle Scholar
Chong, T. T.-L. (2006). The polynomial aggregated AR(1) model. Econometrics J. 9, 98–122.CrossRefGoogle Scholar
Dacunha-Castelle, D. and Fermı´n, L. (2006). Disaggregation of long memory processes on C∞ class. Electron. Commun. Prob. 11, 35–44.CrossRefGoogle Scholar
Dacunha-Castelle, D. and Oppenheim, G. (2001). Mixtures, aggregations and long-memory. Tech. Rep. Université de Paris-Sud.Google Scholar
Dedecker, J. and Doukhan, P. (2003). A new covariance inequality and applications. Stoch. Process. Appl. 106, 63–80.CrossRefGoogle Scholar
Dedecker, J. and Rio, E. (2000). On the functional central limit theorem for stationary processes. Ann. Inst. H. Poincaré Prob. Statist. 36, 1–34.CrossRefGoogle Scholar
Ding, Z. and Granger, C. W. J. (1996). Modeling volatility persistence of speculative returns: a new approach. J. Econometrics 73, 185–215.CrossRefGoogle Scholar
Dobrushin, R. L. and Major, P. (1979). Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Warscheinlichkeitsth. 50, 27–52.CrossRefGoogle Scholar
Giraitis, L. and Surgailis, D. (1985). CLT and other limit theorems for functionals of Gaussian processes. Z. Warscheinlichkeitsth. 70, 191–212.CrossRefGoogle Scholar
Giraitis, L., Leipus, R. and Surgailis, D. (2010). Aggregation of the random coefficient GLARCH(1,1) process. Econometric Theory 26, 406–425.CrossRefGoogle Scholar
Gordin, M. I. (1969). The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188, 739–741.Google Scholar
Granger, C. W. J. (1980). Long memory relationships and the aggregation of dynamic models. J. Econometrics 14, 227–238.CrossRefGoogle Scholar
Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19, 293–325.CrossRefGoogle Scholar
Horváth, L. and Leipus, R. (2009). Effect of aggregation on estimators in AR(1) sequence. Test 18, 546–567.CrossRefGoogle Scholar
Hurd, H. L. and Miamee, A. (2007). Periodically Correlated Random Sequences. Wiley-Interscience, Hoboken, NJ.CrossRefGoogle Scholar
Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes (Fundamental Principals Math. Sci. 288), 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
Jirak, M. (2011). Asymptotic behavior of weakly dependent aggregated processes. Period. Math. Hungar. 62, 39–60.CrossRefGoogle Scholar
Kazakevičius, V., Leipus, R. and Viano, M.-C. (2004). Stability of random coefficient ARCH models and aggregation schemes. J. Econometrics 120, 139–158.CrossRefGoogle Scholar
Koul, H. L. (1992). Weighted Empiricals and Linear Models (IMS Lecture Notes Monogr. Ser. 21). Institute of Mathematical Statistics, Hayward, CA.CrossRefGoogle Scholar
Leipus, R. and Viano, M.-C. (2002). Aggregation in ARCH models. Liet. Mat. Rink. 42, 68–89.Google Scholar
Leipus, R., Oppenheim, G., Philippe, A. and Viano, M.-C. (2006). Orthogonal series density estimation in a disaggregation scheme. J. Statist. Planning Inf. 136, 2547–2571.CrossRefGoogle Scholar
Leonenko, N. N. and Taufer, E. (2005). Convergence of integrated superpositions of Ornstein-Uhlenbeck processes to fractional Brownian motion. Stochastics 77, 477–499.CrossRefGoogle Scholar
Lewbel, A. (1994). Aggregation and simple dynamics. Amer. Econom. Rev. 84, 905–918.Google Scholar
Maxwell, M. and Woodroofe, M. (2000). Central limit theorems for additive functionals of Markov chains. Ann. Prob. 28, 713–724.CrossRefGoogle Scholar
Merlevède, F., Peligrad, M. and Utev, S. (2006). Recent advances in invariance principles for stationary sequences. Prob. Surveys 3, 1–36.CrossRefGoogle Scholar
Oppenheim, G. and Viano, M.-C. (2004). Aggregation of random parameters Ornstein-Uhlenbeck or AR processes: some convergence results. J. Time Ser. Anal. 25, 335–350.CrossRefGoogle Scholar
Peligrad, M. and Utev, S. (2005). A new maximal inequality and invariance principle for stationary sequences. Ann. Prob. 33, 798–815.CrossRefGoogle Scholar
Peligrad, M. and Utev, S. (2006). Central limit theorem for stationary linear processes. Ann. Prob. 34, 1608–1622.CrossRefGoogle Scholar
Peligrad, M. and Utev, S. (2006). Invariance principle for stochastic processes with short memory. In High Dimensional Probability (IMS Lecture Notes Monogr. Ser. 51), Institute of Mathematical Statistics, Beachwood, OH, pp. 18–32.CrossRefGoogle Scholar
Phillips, P. C. B. and Moon, H. R. (1999). Linear regression limit theory for nonstationary panel data. Econometrica 67, 1057–1111.CrossRefGoogle Scholar
Puplinskaitė, D. and Surgailis, D. (2009). Aggregation of random-coefficient AR(1) process with infinite variance and common innovations. Lithuanian Math. J. 49, 446–463.CrossRefGoogle Scholar
Puplinskaitė, D. and Surgailis, D. (2010). Aggregation of a random-coefficient AR(1) process with infinite variance and idiosyncratic innovations. Adv. Appl. Prob. 42, 509–527.CrossRefGoogle Scholar
Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. John Wiley, New York.Google Scholar
Sweeting, T. J. (1977). Speeds of convergence for the multidimensional central limit theorem. Ann. Prob. 5, 28–41.CrossRefGoogle Scholar
Taqqu, M. S. (1974/75). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitstheorie 31, 287–302.CrossRefGoogle Scholar
Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Warscheinlichkeitsth. 50, 53–83.CrossRefGoogle Scholar
Volný, D., Woodroofe, M. and Zhao, O. (2011). Central limit theorems for superlinear processes. Stoch. Dyn. 11, 71–80.CrossRefGoogle Scholar
Wu, W. B. (2005). Nonlinear system theory: another look at dependence. Proc. Natl. Acad. Sci. USA 102, 14150–14154.CrossRefGoogle Scholar
Wu, W. B. (2007). Strong invariance principles for dependent random variables. Ann. Prob. 35, 2294–2320.CrossRefGoogle Scholar
Wu, W. B. and Woodroofe, M. (2004). Martingale approximations for sums of stationary processes. Ann. Prob. 32, 1674–1690.CrossRefGoogle Scholar
Zaffaroni, P. (2004). Contemporaneous aggregation of linear dynamic models in large economies. J. Econometrics 120, 75–102.CrossRefGoogle Scholar
Zaffaroni, P. (2007). Aggregation and memory of models of changing volatility. J. Econometrics 136, 237–249.CrossRefGoogle Scholar
Zaffaroni, P. (2007). Contemporaneous aggregation of GARCH processes. J. Time Ser. Anal. 28, 521–544.CrossRefGoogle Scholar