scholarly journals Functional central limit theorems for certain statistics in an infinite urn scheme

2016 ◽  
Vol 119 ◽  
pp. 344-348 ◽  
Author(s):  
Mikhail Chebunin ◽  
Artyom Kovalevskii
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Urn Scheme ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

scholarly journals Functional Central Limit Theorems for Occupancies and Missing Mass Process in Infinite Urn Models

2020 ◽  
Author(s):  
Mikhail Chebunin ◽  
Sergei Zuyev
Keyword(s):  
Central Limit ◽  
Discrete Time ◽  
Infinite Number ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Urn Models ◽  
Missing Mass ◽  
Urn Scheme ◽  
Functional Central Limit Theorems ◽  
Functional Central Limit

AbstractWe study the infinite urn scheme when the balls are sequentially distributed over an infinite number of urns labeled 1,2,... so that the urn j at every draw gets a ball with probability $$p_j$$ p j , where $$\sum _j p_j=1$$ ∑ j p j = 1 . We prove functional central limit theorems for discrete time and the Poissonized version for the urn occupancies process, for the odd occupancy and for the missing mass processes extending the known non-functional central limit theorems.

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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