The asymptotic theory of linear time-series models

1973 ◽  
Vol 10 (1) ◽  
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.

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.

scholarly journals Central limit theorem for linear processes generated by IID random variables under the sub-linear expectation

2021 ◽  
Vol 36 (2) ◽  
pp. 243-255
Author(s):  
Wei Liu ◽  
Yong Zhang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Invariance Principle ◽  
Random Variables ◽  
Probability Measures ◽  
Linear Processes ◽  
The Central Limit Theorem

AbstractIn this paper, we investigate the central limit theorem and the invariance principle for linear processes generated by a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng [19]. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov’s central limit theorem and invariance principle to the case where probability measures are no longer additive.

scholarly journals Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space?

2021 ◽  
Author(s):  
Felix Herold ◽  
Daniel Hug ◽  
Christoph Thäle
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Hyperbolic Space ◽  
Central Limit ◽  
Hausdorff Measure ◽  
Rates Of Convergence ◽  
Limit Problem ◽  
Dimensional Hausdorff Measure ◽  
The Central Limit Theorem ◽  
Order Quantities

AbstractPoisson processes in the space of $$(d-1)$$ ( d - 1 ) -dimensional totally geodesic subspaces (hyperplanes) in a d-dimensional hyperbolic space of constant curvature $$-1$$ - 1 are studied. The k-dimensional Hausdorff measure of their k-skeleton is considered. Explicit formulas for first- and second-order quantities restricted to bounded observation windows are obtained. The central limit problem for the k-dimensional Hausdorff measure of the k-skeleton is approached in two different set-ups: (i) for a fixed window and growing intensities, and (ii) for fixed intensity and growing spherical windows. While in case (i) the central limit theorem is valid for all $$d\ge 2$$ d ≥ 2 , it is shown that in case (ii) the central limit theorem holds for $$d\in \{2,3\}$$ d ∈ { 2 , 3 } and fails if $$d\ge 4$$ d ≥ 4 and $$k=d-1$$ k = d - 1 or if $$d\ge 7$$ d ≥ 7 and for general k. Also rates of convergence are studied and multivariate central limit theorems are obtained. Moreover, the situation in which the intensity and the spherical window are growing simultaneously is discussed. In the background are the Malliavin–Stein method for normal approximation and the combinatorial moment structure of Poisson U-statistics as well as tools from hyperbolic integral geometry.

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