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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Bollerslev T (1986). Generalized autoregressive conditional heteroskedasticity. J Econometrics 31: 307–327
Doukhan P (1994) Mixing: properties and examples. Lecture Notes in Statistics 85. Springer-Verlag
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
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
Doukhan P and Louhichi S (1999). A new weak dependence condition and applications to moment inequalities. Stoch Proc Appl 84: 313–342
Doukhan P, Wintenberger O (2007) An invariance principle for weakly dependent stationary general models. Probab Math Stat 27:45–73
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)
Doukhan P, Madre H, Rosenbaum M (2007) Weak dependence for infinite ARCH-type bilinear models. Statistics 41:31–45
Engle RF (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50: 987–1007
Giraitis L, Leipus R, Surgailis D (2005) Recent advances in ARCH modelling. In: Teyssière G, Kirman A (eds) Long-memory in economics. Springer Verlag
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
Giraitis L and Surgailis D (2002). ARCH-type bilinear models with double long memory. Stoch Proc Appl 100: 275–300
Horvath L and Shao Q-M (1999). Limit theorems for quadratic forms with applications to Whittle’s estimate. Ann Appl Probab 9: 146–187
Leonov VP and Shiryaev AN (1959). On a method of semi-invariants. Theor Probab Appl 4: 319–329
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
Rio E (1996). Sur le théorème de berry-esseen pour les suites faiblement dépendantes. Probab Th Rel Fields 104: 255–282
Robinson PM (1991) Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. J Econometrics 47:67–84
Rosenblatt M (1985). Stationary processes and random fields. Birkhäuser, Boston
Rosenblatt M (2000) Gaussian and non-Gaussian linear time series and random fields. Springer Series in Statistics. Springer-Verlag, New York
C. José Rafael León—Partially supported by the program ECOS-NORD of Fonacit, Venezuela.
About this article
Cite this article
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
- Central limit theorem
- Weakly dependent processes
- Sample moments and cumulants
Mathematics Subject Classification