STATIONARITY AND CENTRAL LIMIT THEOREM ASSOCIATED WITH BILINEAR TIME SERIES MODELS

1991 ◽  
Vol 12 (4) ◽  
pp. 301-313 ◽  
Author(s):  
Kamal C. Chanda
Keyword(s):  
Time Series ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Time Series Models

The asymptotic theory of linear time-series models

1973 ◽  
Vol 10 (01) ◽  
pp. 130-145 ◽  
Author(s):  
E. J. Hannan
Keyword(s):  
Time Series ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Theory ◽  
Linear Time ◽  
Stationary Time Series ◽  
Linear Predictor ◽  
The Central Limit Theorem ◽  
Best Linear Predictor

A linear time-series model is considered to be one for which a stationary time series, which is purely non-deterministic, has the best linear predictor equal to the best predictor. A general inferential theory is constructed for such models and various estimation procedures are shown to be equivalent. The treatment is considerably more general than previous treatments. The case where the series has mean which is a linear function of very general kinds of regressor variables is also discussed and a rather general form of central limit theorem for regression is proved. The central limit results depend upon forms of the central limit theorem for martingales.

CENTRAL LIMIT THEOREM FOR THE LOG-REGRESSION WAVELET ESTIMATION OF THE MEMORY PARAMETER IN THE GAUSSIAN SEMI-PARAMETRIC CONTEXT

Fractals ◽  
2007 ◽  
Vol 15 (04) ◽  
pp. 301-313 ◽  
Author(s):  
E. MOULINES ◽  
F. ROUEFF ◽  
MURAD S. TAQQU
Keyword(s):  
Time Series ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Variance ◽  
Spectral Density Function ◽  
Dependence Structure ◽  
Asymptotically Normal ◽  
The One ◽  
Gaussian Time

We consider a Gaussian time series, stationary or not, with long memory exponent d ∈ ℝ. The generalized spectral density function of the time series is characterized by d and by a function f*(λ) which specifies the short-range dependence structure. Our setting is semi-parametric in that both d and f* are unknown, and only the smoothness of f* around λ = 0 matters. The parameter d is the one of interest. It is estimated by regression using the wavelet coefficients of the time series, which are dependent when d ≠ 0. We establish a Central Limit Theorem (CLT) for the resulting estimator [Formula: see text]. We show that the deviation [Formula: see text], adequately normalized, is asymptotically normal and specify the asymptotic variance.

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