New Ways to Prove Central Limit Theorems

1985 ◽  
Vol 1 (3) ◽  
pp. 295-313 ◽  
Author(s):  
David Pollard
Keyword(s):  
Maximum Likelihood ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Limit Theorems ◽  
Maximum Likelihood Estimators ◽  
Empirical Processes ◽  
Central Limit Theorems ◽  
Criterion Function ◽  
Nonstandard Conditions

This paper describes some techniques for proving asymptotic normality of statistics defined by maximization of random criterion function. The techniques are based on a combination of recent results from the theory of empirical processes and a method of Huber for the study of maximum likelihood estimators under nonstandard conditions.

Functional and random central limit theorems for the Robbins-Munro process

1976 ◽  
Vol 13 (1) ◽  
pp. 148-154 ◽  
Author(s):  
D. L. McLeish
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Functional Central Limit Theorem ◽  
Stopping Rules ◽  
The Central Limit Theorem ◽  
Functional Central Limit

A functional central limit theorem extending the central limit theorem of Chung (1954) for the Robbins–Munro procedure is proved. It is shown that the asymptotic normality is preserved under certain random stopping rules.

scholarly journals Central limit theorems for local empirical processes near boundaries of sets

Bernoulli ◽  
2011 ◽  
Vol 17 (2) ◽  
pp. 545-561 ◽  
Author(s):  
John H.J. Einmahl ◽  
Estáte V. Khmaladze
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Empirical Processes ◽  
Central Limit Theorems

Functional and random central limit theorems for the Robbins-Munro process

1976 ◽  
Vol 13 (01) ◽  
pp. 148-154
Author(s):  
D. L. McLeish
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Functional Central Limit Theorem ◽  
Stopping Rules ◽  
The Central Limit Theorem ◽  
Functional Central Limit

A functional central limit theorem extending the central limit theorem of Chung (1954) for the Robbins–Munro procedure is proved. It is shown that the asymptotic normality is preserved under certain random stopping rules.

Asymptotic Normality of some Estimators

1981 ◽  
Vol 30 (1-2) ◽  
pp. 13-22
Author(s):  
Adnan M. Awad
Keyword(s):  
Central Limit Theorem ◽  
Maximum Likelihood ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Central Limit ◽  
Continuous Mapping ◽  
Maximum Likelihood Estimators ◽  
Posterior Distributions ◽  
Martingale Central Limit Theorem ◽  
Log Likelihood

This paper uses martingale central limit theorem and continuous mapping theorem to establish asymptotic normality of log-likelihood ratio process, maximum likelihood estimators and the posterior distributions. Illustrative examples are given.

scholarly journals Pressure Function and Limit Theorems for Almost Anosov Flows

2021 ◽  
Vol 382 (1) ◽  
pp. 1-47
Author(s):  
Henk Bruin ◽  
Dalia Terhesiu ◽  
Mike Todd
Keyword(s):  
Thermodynamic Properties ◽  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Stable Laws ◽  
Analytic Framework ◽  
Pressure Function ◽  
Hyperbolic Flows ◽  
Anosov Flows ◽  
Almost Anosov

AbstractWe obtain limit theorems (Stable Laws and Central Limit Theorems, both standard and non-standard) and thermodynamic properties for a class of non-uniformly hyperbolic flows: almost Anosov flows, constructed here. The link between the pressure function and limit theorems is studied in an abstract functional analytic framework, which may be applicable to other classes of non-uniformly hyperbolic flows.

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