scholarly journals Almost sure invariance principle for some maps of an interval

1985 ◽  
Vol 5 (4) ◽  
pp. 625-640 ◽  
Author(s):  
Krystyna Ziemian
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Invariance Principle ◽  
Critical Points ◽  
Measurable Function ◽  
Schwarzian Derivative ◽  
Almost Sure Invariance Principle

AbstractWe prove an almost sure invariance principle and a central limit theorem for the process , where f is a map of an interval with a non-positive Schwarzian derivative whose trajectories of critical points stay far from the critical points, and F is a measurable function with bounded p-variation (p ≥ 1).The almost sure invariance principle implies the Log-log laws, integral tests and a distributional type of invariance principle for the process .

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.

EMPIRICAL INVARIANCE PRINCIPLE FOR ERGODIC TORUS AUTOMORPHISMS: GENERICITY

2008 ◽  
Vol 08 (02) ◽  
pp. 173-195 ◽  
Author(s):  
OLIVIER DURIEU ◽  
PHILIPPE JOUAN
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Invariance Principle ◽  
Morse Function ◽  
Empirical Process ◽  
Stationary Process ◽  
Sufficient Condition ◽  
Hölder Continuous ◽  
Morse Functions

We consider the dynamical system given by an algebraic ergodic automorphism T on a torus. We study a Central Limit Theorem for the empirical process associated to the stationary process (f◦Ti)i∈ℕ, where f is a given ℝ-valued function. We give a sufficient condition on f for this Central Limit Theorem to hold. In the second part, we prove that the distribution function of a Morse function is continuously differentiable if the dimension of the manifold is at least three and Hölder continuous if the dimension is one or two. As a consequence, the Morse functions satisfy the empirical invariance principle, which is therefore generically verified.

Asymptotic inference for an ising lattice. II

1977 ◽  
Vol 9 (3) ◽  
pp. 476-501 ◽  
Author(s):  
D. K. Pickard
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Ising Model ◽  
Central Limit ◽  
Critical Points ◽  
Limit Theorems ◽  
Relative Rate ◽  
Ising Lattice ◽  
The Central Limit Theorem ◽  
Significant Case

In Pickard (1976) limit theorems were obtained for the classical Ising model at non-critical points. These determined the asymptotic distribution of the sample nearest-neighbour correlation, thereby providing a basis for statistical inference by confidence intervals. In this paper, these limit theorems are extended to the statistically significant case of different vertical and horizontal interactions. Results at critical points are also obtained. Critical points clearly have the potential to seriously distort statistical inferences, especially in their immediate neighbourhoods. For our Ising model it turns out that such distortion is relatively minor. Surprisingly, in the two-parameter case the correlation between the sufficient statistics exhibits peculiar asymptotic behaviour resulting in a singular covariance matrix at critical points in the central limit theorem. Finally, at critical points, unusual norming constants are required for the central limit theorem, and our results are much more sensitive to the relative rate at which m, n tend to infinity.

Asymptotic inference for an ising lattice. II

1977 ◽  
Vol 9 (03) ◽  
pp. 476-501 ◽  
Author(s):  
D. K. Pickard
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Ising Model ◽  
Central Limit ◽  
Critical Points ◽  
Limit Theorems ◽  
Relative Rate ◽  
Ising Lattice ◽  
The Central Limit Theorem ◽  
Significant Case

In Pickard (1976) limit theorems were obtained for the classical Ising model at non-critical points. These determined the asymptotic distribution of the sample nearest-neighbour correlation, thereby providing a basis for statistical inference by confidence intervals. In this paper, these limit theorems are extended to the statistically significant case of different vertical and horizontal interactions. Results at critical points are also obtained. Critical points clearly have the potential to seriously distort statistical inferences, especially in their immediate neighbourhoods. For our Ising model it turns out that such distortion is relatively minor. Surprisingly, in the two-parameter case the correlation between the sufficient statistics exhibits peculiar asymptotic behaviour resulting in a singular covariance matrix at critical points in the central limit theorem. Finally, at critical points, unusual norming constants are required for the central limit theorem, and our results are much more sensitive to the relative rate at which m, n tend to infinity.

Sign in / Sign up

Export Citation Format

Share Document