scholarly journals Functional Central Limit Theorems and P(ϕ)1-Processes for the Relativistic and Non-Relativistic Nelson Models

2020 ◽  
Vol 23 (2) ◽  
Author(s):  
Soumaya Gheryani ◽  
Fumio Hiroshima ◽  
József Lőrinczi ◽  
Achref Majid ◽  
Habib Ouerdiane
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

CENTRAL LIMIT AND FUNCTIONAL CENTRAL LIMIT THEOREMS FOR HILBERT-VALUED DEPENDENT HETEROGENEOUS ARRAYS WITH APPLICATIONS

1998 ◽  
Vol 14 (2) ◽  
pp. 260-284 ◽  
Author(s):  
Xiaohong Chen ◽  
Halbert White
Keyword(s):  
Central Limit ◽  
Adaptive Learning ◽  
Limit Theorems ◽  
Asymptotic Distributions ◽  
Central Limit Theorems ◽  
Cumulative Distribution ◽  
Density Estimator ◽  
Dependent Observations ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

We obtain new central limit theorems (CLT's) and functional central limit theorems (FCLT's) for Hilbert-valued arrays near epoch dependent on mixing processes, and also new FCLT's for general Hilbert-valued adapted dependent heterogeneous arrays. These theorems are useful in delivering asymptotic distributions for parametric and nonparametric estimators and their functionals in time series econometrics. We give three significant applications for near epoch dependent observations: (1) A new CLT for any plug-in estimator of a cumulative distribution function (c.d.f.) (e.g., an empirical c.d.f., or a c.d.f. estimator based on a kernel density estimator), which can in turn deliver distribution results for many Von Mises functionals; (2) a new limiting distribution result for degenerate U-statistics, which delivers distribution results for Bierens's integrated conditional moment tests; (3) a new functional central limit result for Hilbert-valued stochastic approximation procedures, which delivers distribution results for nonparametric recursive generalized method of moment estimators, including nonparametric adaptive learning models.

Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach

1973 ◽  
Vol 5 (01) ◽  
pp. 119-137 ◽  
Author(s):  
D. J. Scott
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Functional Central Limit Theorem ◽  
Stationary Increments ◽  
Mixing Sequences ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

The Skorokhod representation for martingales is used to obtain a functional central limit theorem (or invariance principle) for martingales. It is clear from the method of proof that this result may in fact be extended to the case of triangular arrays in which each row is a martingale sequence and the second main result is a functional central limit theorem for such arrays. These results are then used to obtain two functional central limit theorems for processes with stationary ergodic increments following on from the work of Gordin. The first of these theorems extends a result of Billingsley for Φ-mixing sequences.

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