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A functional limit theorem for η-weakly dependent processes and its applications

Abstract

We prove a general functional central limit theorem for weak dependent time series. A very large variety of models, for instance, causal or non causal linear, ARCH(∞), LARCH(∞), Volterra processes, satisfies this theorem. Moreover, it provides numerous applications as well for bounding the distance between the empirical mean and the Gaussian measure than for obtaining central limit theorem for sample moments and cumulants.

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References

  1. Bollerslev T (1986). Generalized autoregressive conditional heteroskedasticity. J Econometrics 31: 307–327

  2. Doukhan P (1994) Mixing: properties and examples. Lecture Notes in Statistics 85. Springer-Verlag

  3. Doukhan P (2003) Models inequalities and limit theorems for stationary sequences. In: Doukhan et al. (eds) Theory and applications of long range dependence. Birkhäuser, pp 43–101

  4. Doukhan P and Lang G (2002). Rates in the empirical central limit theorem for stationary weakly dependent random fields. Stat Inference Stoch Process 5: 199–228

  5. Doukhan P and Louhichi S (1999). A new weak dependence condition and applications to moment inequalities. Stoch Proc Appl 84: 313–342

  6. Doukhan P, Wintenberger O (2007) An invariance principle for weakly dependent stationary general models. Probab Math Stat 27:45–73

  7. Doukhan P, Teyssiere G, Winant P (2006) Vector valued ARCH(∞) processes. In: Bertail P, Doukhan P, Soulier P (eds) Lecture Note in Statistics, 187. Special issue on time series (to appear)

  8. Doukhan P, Madre H, Rosenbaum M (2007) Weak dependence for infinite ARCH-type bilinear models. Statistics 41:31–45

  9. Engle RF (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50: 987–1007

  10. Giraitis L, Leipus R, Surgailis D (2005) Recent advances in ARCH modelling. In: Teyssière G, Kirman A (eds) Long-memory in economics. Springer Verlag

  11. Giraitis L and Surgailis D (1990). A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle’s estimate. Probab Th Rel Fields 86: 87–104

  12. Giraitis L and Surgailis D (2002). ARCH-type bilinear models with double long memory. Stoch Proc Appl 100: 275–300

  13. Horvath L and Shao Q-M (1999). Limit theorems for quadratic forms with applications to Whittle’s estimate. Ann Appl Probab 9: 146–187

  14. Leonov VP and Shiryaev AN (1959). On a method of semi-invariants. Theor Probab Appl 4: 319–329

  15. Pène F (2005). Rate of convergence in the multidimensional CLT for stationary processes. Application to the Knudsen gas and to the Sinai billiard. Ann Appl Probab 15: 2331–2392

  16. Rio E (1996). Sur le théorème de berry-esseen pour les suites faiblement dépendantes. Probab Th Rel Fields 104: 255–282

  17. Robinson PM (1991) Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. J Econometrics 47:67–84

  18. Rosenblatt M (1985). Stationary processes and random fields. Birkhäuser, Boston

  19. Rosenblatt M (2000) Gaussian and non-Gaussian linear time series and random fields. Springer Series in Statistics. Springer-Verlag, New York

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Author information

Correspondence to Jean-Marc Bardet.

Additional information

C. José Rafael León—Partially supported by the program ECOS-NORD of Fonacit, Venezuela.

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Bardet, J., Doukhan, P. & León, J.R. A functional limit theorem for η-weakly dependent processes and its applications. Stat Infer Stoch Process 11, 265–280 (2008). https://doi.org/10.1007/s11203-007-9015-y

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Keywords

  • Central limit theorem
  • Weakly dependent processes
  • Sample moments and cumulants

Mathematics Subject Classification

  • 60F05
  • 62F12
  • 62M10