Skip to main content
SpringerLink
Log in

Asymptotic behavior of weakly dependent aggregated processes

  • Published:
Periodica Mathematica Hungarica Aims and scope Submit manuscript

Abstract

Aggregated processes appear in many areas of statistics, natural sciences and economics and studying their behavior has a considerable importance from a purely probabilistic point of view as well. Granger (1980) showed that aggregating processes of simple structure can lead to processes with much more complex dynamics, in particular, aggregating random coefficient AR(1) processes can result in long memory processes. This opens a new way to analyze complex processes by constructing such processes from simple ‘building blocks’ via aggregation.

The basic statistical problem of aggregation theory is, given a sample {Y (N)1 , …, Y (N) n } of size n of the N-fold aggregated process, to draw conclusions for the structure of the constituting processes (“disaggregation”) and use this for describing the asymptotic behavior of the aggregated process. Probabilistically, this requires determining the limit distribution of nonlinear functionals of {Y (N)1 , …, Y (N) n }, which depends sensitively on the relative order of n and N.

In this survey paper, we give a detailed asymptotic study of aggregated linear processes with an arbitrary (possibly infinite) number of parameters and apply the results to the disaggregation problem of AR(1) and AR(2) processes. We also discuss the problem of long memory of aggregated processes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Abramowitz and I. A. Stegun (Editors), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover, 1992.

  2. D. W. K. Andrews, Heteroskedasticity and autocorrelation consistent covariance matrix estimation, Technical report, 1988.

  3. D. D. Ang, R. Gorenflo, V. K. Le and D. D. Trong, Moment theory and some inverse problems in potential theory and heat conduction, Lecture Notes in Mathematics 1792, Springer, 2002.

  4. D. D. Ang, van Nhan Nguyen and Dinh Ngoc Thanh, A nonlinear integral equation of gravimetry: uniqueness and approximation by linear moments, Vietnam J. Math., 27 (1999), 61–67.

    MATH  MathSciNet  Google Scholar 

  5. V. V. Anh, C. C. Heyde and N. N. Leonenko, Dynamic models of long-memory processes driven by Lévy noise, J. Appl. Probab., 39 (2002), 730–747.

    Article  MATH  MathSciNet  Google Scholar 

  6. V. V. Anh, V. P. Knopova and N. N. Leonenko, Continuous-time stochastic processes with cyclical long-range dependence, Aust. N. Z. J. Stat., 46 (2004), 275–296.

    Article  MATH  MathSciNet  Google Scholar 

  7. V. V. Anh, N. N. Leonenko and R. Mcvinish, Models for fractional Riesz-Bessel motion and related processes, Fractals, 9 (2001), 329–346.

    Article  Google Scholar 

  8. M. A. Arcones, Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors, Ann. Probab., 22 (1994), 2242–2274.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. A. Arcones, Distributional limit theorems over a stationary Gaussian sequence of random vectors, Stochastic Process. Appl., 88 (2000), 135–159.

    Article  MATH  MathSciNet  Google Scholar 

  10. R. T. Baillie, T. Bollerslev and H. O. Mikkelsen, Fractionally integrated generalized autoregressive conditional heteroskedasticity, J. Econometrics, 74 (1996), 3–30.

    Article  MATH  MathSciNet  Google Scholar 

  11. J-M. Bardet, G. Lang, G. Oppenheim, A. Philippe and M. S. Taqqu, Generators of long-range dependent processes: a survey, Theory and applications of long-range dependence, Birkhäuser, Boston, 2003, 579–623.

    Google Scholar 

  12. O. E. Barndorff-Nielsen, Superposition of Ornstein-Uhlenbeck type processes, Teor. Veroyatnost. i Primenen., 45 (2000), 289–311.

    MathSciNet  Google Scholar 

  13. O. E. Barndorff-Nielsen and N. N. Leonenko, Spectral properties of superpositions of Ornstein-Uhlenbeck type processes, Methodol. Comput. Appl. Probab., 7 (2005), 335–352.

    Article  MATH  MathSciNet  Google Scholar 

  14. J. Beran, S. Ghosh and M. Schützner, From short to long memory: Aggregation and estimation, Computational Statistics and Data Analysis, to appear.

  15. J. Beran and D. Ocker, Temporal aggregation of stationary and nonstationary farima (p, d, 0)-models, CoFE Discussion Paper, University of Konstanz, 2000.

  16. P. J. Brockwell and R. A. Davis, Time series: theory and methods, Springer Series in Statistics, Springer, 1991.

    Book  Google Scholar 

  17. D. Celov, R. Leipus and A. Philippe, Asymptotic normality of the mixture density estimator in a disaggregation scheme, J. Nonparametr. Stat., to appear.

  18. D. Celov, R. Leipus and A. Philippe, Time series aggregation, disaggregation, and long memory, Liet. Mat. Rink., 47 (2007), 466–481.

    MathSciNet  Google Scholar 

  19. D. Dacunha-Castelle and L. Fermín, Disaggregation of long memory processes on C∞ class, Electr. Comm. Probab., 11 (2006), 35–44.

    MATH  Google Scholar 

  20. D. Dacunha-Castelle and G. Oppenheim, Mixtures, aggregations and long-memory, Technical report, Université de Paris Sud., 2001.

  21. J. Dedecker and P. Doukhan, A new covariance inequality and applications, Stochastic Process. Appl., 106 (2003), 63–80.

    Article  MATH  MathSciNet  Google Scholar 

  22. J. Dedecker and E. Rio, On the functional central limit theorem for stationary processes, Ann. Inst. H. Poincaré Probab. Statist., 36 (2000), 1–34.

    Article  MATH  MathSciNet  Google Scholar 

  23. Z. Ding and C. W. J. Granger, Modeling volatility persistence of speculative returns: a new approach, J. Econometrics, 73 (1996), 185–215.

    Article  MATH  MathSciNet  Google Scholar 

  24. R. L. Dobrushin and P. Major, Non-central limit theorems for nonlinear functionals of Gaussian fields, Z. Wahrsch. Verw. Gebiete, 50 (1979), 27–52.

    Article  MATH  MathSciNet  Google Scholar 

  25. W. Feller, An introduction to probability theory and its applications, Vol. II, Wiley, 1971.

  26. J. Geweke and S. Porter-Hudak, The estimation and application of long memory time series models, J. Time Ser. Anal., 4 (1983), 221–238.

    Article  MATH  MathSciNet  Google Scholar 

  27. L. Giraitis, P. Kokoszka, R. Leipus and G. Teyssière, Rescaled variance and related tests for long memory in volatility and levels, J. Econometrics, 112 (2003), 265–294.

    Article  MATH  MathSciNet  Google Scholar 

  28. L. Giraitis, R. Leipus and D. Surgailis, Aggregation of the random coefficient GLARCH(1, 1) process, Econometric Theory, 26 (2010), 406–425.

    Article  MATH  MathSciNet  Google Scholar 

  29. L. Giraitis, P. M. Robinson and D. Surgailis, A model for long memory conditional heteroscedasticity, Ann. Appl. Probab., 10 (2000), 1002–1024.

    Article  MATH  MathSciNet  Google Scholar 

  30. L. Giraitis and D. Surgailis, CLT and other limit theorems for functionals of Gaussian processes, Z. Wahrsch. Verw. Gebiete, 70 (1985), 191–212.

    Article  MATH  MathSciNet  Google Scholar 

  31. A. Goldenshluger and V. Spokoiny, On the shape-from-moments problem and recovering edges from noisy Radon data, Probab. Theory Related Fields, 128 (2004), 123–140.

    Article  MATH  MathSciNet  Google Scholar 

  32. M. I. Gordin, The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR, 188 (1969), 739–741.

    MathSciNet  Google Scholar 

  33. C. W. J. Granger, Long memory relationships and the aggregation of dynamic models, J. Econometrics, 14 (1980), 227–238.

    Article  MATH  MathSciNet  Google Scholar 

  34. M. Haye, G. Oppenheim and M.-C. Viano, Long memory with seasonal effects, Stat. Inference Stoch. Process., 3 (2000), 53–68.

    Article  MATH  MathSciNet  Google Scholar 

  35. L. Horváth and R. Leipus, Effect of aggregation on estimators in AR(1) sequence, Test, 136 (2006), 2547–2571.

    Google Scholar 

  36. H. Hurst, Long term storage capacity of reservoirs, Rev. Econom. Stud., Transactions of the American Society of Civil Engineers, 116 (1951), 770799.

    Google Scholar 

  37. J. Jirak, Limit theorems for aggregated processes, to appear.

  38. J. Jirak, Aggregation and disaggregation of AR(1) processes and moment based density estimation, preprint.

  39. J. Jirak, On the aggregation and disaggregation of AR(1) and AR(2) processes, preprint.

  40. V. Kazakevicius, R. Leipus and M.-C. Viano, Stability of random coefficient ARCH models and aggregation schemes, J. Econometrics, 120 (2004), 139–158.

    Article  MathSciNet  Google Scholar 

  41. D. Kwiatkowski, P. C. B. Phillips and P. Schmidt, Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root?, Cowles Foundation Discussion Papers 979, Cowles Foundation, Yale University, 1991.

  42. R. Leipus, G. Oppenheim, A. Philippe and M-C. Viano, Orthogonal series density estimation in a disaggregation scheme, J. Statist. Plann. Inference, 136 (2006), 2547–2571.

    Article  MATH  MathSciNet  Google Scholar 

  43. R. Leipus and M.-C. Viano, Aggregation in ARCH models, Liet. Mat. Rink., 42 (2002), 68–89.

    MathSciNet  Google Scholar 

  44. N. N. Leonenko and E. Taufer, Convergence of integrated superpositions of Ornstein-Uhlenbeck processes to fractional Brownian motion, Stochastics, 77 (2005), 477–499.

    MATH  MathSciNet  Google Scholar 

  45. A. W. Lo, Long-term memory in stock market prices, Econometrica, 59 (1991), 1279–313.

    Article  MATH  Google Scholar 

  46. I. N. Lobato and P. M. Robinson, A nonparametric test for I(0), Rev. Econom. Stud., 65 (1998), 475–495.

    Article  MATH  MathSciNet  Google Scholar 

  47. B. B. Mandelbrot and M. S. Taqqu, Robust R/S analysis of long-run serial correlation, Proceedings of the 42nd session of the International Statistical Institute, Vol. 2 (Manila, 1979), Bull. Inst. Internat. Statist., 48 (1979), 69–99.

    MATH  MathSciNet  Google Scholar 

  48. M. Maxwell and M. Woodroofe, Central limit theorems for additive functionals of Markov chains, Ann. Probab., 28 (2000), 713–724.

    Article  MATH  MathSciNet  Google Scholar 

  49. F. Merlevède, M. Peligrad and S. Utev, Recent advances in invariance principles for stationary sequences, Probab. Surv., 3 (2006), 1–36.

    Article  MATH  MathSciNet  Google Scholar 

  50. R. Mnatsakanov and F. H. Ruymgaart, Some results for moment-empirical cumulative distribution functions, J. Nonparametr. Stat., 17 (2005), 733–744.

    Article  MATH  MathSciNet  Google Scholar 

  51. R. M. Mnatsakanov, Hausdorff moment problem: Reconstruction of probability density functions, Statist. Probab. Lett., 78 (2008), 1869–1877.

    Article  MATH  MathSciNet  Google Scholar 

  52. H. R. Moon and P. C. B. Phillips, Linear regression limit theory for nonstationary panel data, Econometrica, 67 (1999), 1057–1111.

    Article  MATH  MathSciNet  Google Scholar 

  53. T. M. Pham Ngoc, A statistical minimax approach to the Hausdorff moment problem, Inverse Problems, 24 (2008), 045018-13.

    MathSciNet  Google Scholar 

  54. G. Oppenheim and M.-C. Viano, Obtaining seasonal long-memory by aggregating simple discrete or continuous time random coefficients short memory processes, Pub. IRMA Lille, 49 (1999), V.

    Google Scholar 

  55. G. Oppenheim and M.-C. Viano, Aggregation of random parameters Ornstein-Uhlenbeck or AR processes: some convergence results, J. Time Ser. Anal., 25 (2004), 335–350.

    Article  MATH  MathSciNet  Google Scholar 

  56. M. Peligrad and S. Utev, A new maximal inequality and invariance principle for stationary sequences, Ann. Probab., 33 (2005), 798–815.

    Article  MATH  MathSciNet  Google Scholar 

  57. M. Peligrad and S. Utev, Central limit theorem for stationary linear processes, Ann. Probab., 34 (2006), 1608–1622.

    Article  MATH  MathSciNet  Google Scholar 

  58. M. Peligrad and S. Utev, Invariance principle for stochastic processes with short memory, High dimensional probability, IMS Lecture Notes Monogr. Ser. 51, Inst. Math. Statist., Beachwood, OH, 2006, 18–32.

    Chapter  Google Scholar 

  59. W. Philipp and W.F. Stout, Almost sure invariance principles for partial sums of weakly dependent random variables, Memoirs of the AMS, No. 161, Amer. Math. Soc., 1975.

  60. P. M. Robinson and P. Zaffaroni, Nonlinear time series with long memory: a model for stochastic volatility, J. Statist. Plann. Inference, 68 (1998), 359–371.

    Article  MATH  MathSciNet  Google Scholar 

  61. D. Surgailis, Zones of attraction of self-similar multiple integrals, Lithuanian Math. J., 22 (1982), 327–340.

    Article  MathSciNet  Google Scholar 

  62. M. S. Taqqu, Weak convergence to fractional Brownian motion and to the Rosenblatt process, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 31 (1974/75), 287–302.

    Article  MathSciNet  Google Scholar 

  63. M. S. Taqqu, Convergence of integrated processes of arbitrary Hermite rank, Z. Wahrsch. Verw. Gebiete, 50 (1979), 53–83.

    Article  MATH  MathSciNet  Google Scholar 

  64. M. Woodroofe and W. B. Wu, Martingale approximations for sums of stationary processes, Ann. Probab., 32 (2004), 1674–1690.

    Article  MATH  MathSciNet  Google Scholar 

  65. P. Zaffaroni, Contemporaneous aggregation of linear dynamic models in large economies, J. Econometrics, 120 (2004), 75–102.

    Article  MathSciNet  Google Scholar 

  66. P. Zaffaroni, Aggregation and memory of models of changing volatility, J. Econometrics, 136 (2007), 237–249.

    Article  MathSciNet  Google Scholar 

  67. P. Zaffaroni, Contemporaneous aggregation of GARCH processes, J. Time Ser. Anal., 28 (2007), 521–544.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Statistik, Technische Universität Graz, Münzgrabenstraße 11, 8010, Graz, Austria

    Moritz Jirak

Authors
  1. Moritz Jirak
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Moritz Jirak.

Additional information

Dedicated to Endre Csáki and Pál Révész on the occasion of their 75th birthdays

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jirak, M. Asymptotic behavior of weakly dependent aggregated processes. Period Math Hung 62, 39–60 (2011). https://doi.org/10.1007/s10998-011-5039-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10998-011-5039-6

Mathematics subject classification number

Key words and phrases

Search

Navigation