A joint central limit theorem for the sample mean and regenerative variance estimator

1987 ◽  
Vol 8 (1) ◽  
pp. 41-55 ◽  
Author(s):  
P. W. Glynn ◽  
D. L. Iglehart
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Variance Estimator ◽  
Sample Mean

scholarly journals Counterexamples to the classical central limit theorem for triplewise independent random variables having a common arbitrary margin

2021 ◽  
Vol 9 (1) ◽  
pp. 424-438
Author(s):  
Guillaume Boglioni Beaulieu ◽  
Pierre Lafaye de Micheaux ◽  
Frédéric Ouimet
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Marginal Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Sample Mean ◽  
Non Gaussian ◽  
Mutually Independent ◽  
Specific Sequences

Abstract We present a general methodology to construct triplewise independent sequences of random variables having a common but arbitrary marginal distribution F (satisfying very mild conditions). For two specific sequences, we obtain in closed form the asymptotic distribution of the sample mean. It is non-Gaussian (and depends on the specific choice of F). This allows us to illustrate the extent of the ‘failure’ of the classical central limit theorem (CLT) under triplewise independence. Our methodology is simple and can also be used to create, for any integer K, new K-tuplewise independent sequences that are not mutually independent. For K [four.tf], it appears that the sequences created using our methodology do verify a CLT, and we explain heuristically why this is the case.

Central Limit Theorem for Absolute Deviations from the Sample Mean and Applications

1979 ◽  
Vol 22 (4) ◽  
pp. 391-396
Author(s):  
D. L. McLeish
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variables ◽  
Frequent Occurrence ◽  
Sample Mean ◽  
Independent Identically Distributed

The following type of argument is rendered almost believable by its frequent occurrence in elementary courses in statistics. Let ξi be a sequence of independent identically distributed random variables with means μ variances σ2.

scholarly journals The Correct Asymptotic Variance for the Sample Mean of a Homogeneous Poisson Marked Point Process

2013 ◽  
Vol 50 (03) ◽  
pp. 889-892
Author(s):  
William Garner ◽  
Dimitris N. Politis
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Point Process ◽  
Central Limit ◽  
Asymptotic Variance ◽  
Marked Point ◽  
Dimensional Case ◽  
Marked Point Process ◽  
Sample Mean ◽  
Correct Expression

The asymptotic variance of the sample mean of a homogeneous Poisson marked point process has been studied in the literature, but confusion has arisen as to the correct expression due to some technical intricacies. This note sets the record straight with regards to the variance of the sample mean. In addition, a central limit theorem in the general d-dimensional case is also established.

scholarly journals The Correct Asymptotic Variance for the Sample Mean of a Homogeneous Poisson Marked Point Process

2013 ◽  
Vol 50 (3) ◽  
pp. 889-892
Author(s):  
William Garner ◽  
Dimitris N. Politis
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Point Process ◽  
Central Limit ◽  
Asymptotic Variance ◽  
Marked Point ◽  
Dimensional Case ◽  
Marked Point Process ◽  
Sample Mean ◽  
Correct Expression

The asymptotic variance of the sample mean of a homogeneous Poisson marked point process has been studied in the literature, but confusion has arisen as to the correct expression due to some technical intricacies. This note sets the record straight with regards to the variance of the sample mean. In addition, a central limit theorem in the general d-dimensional case is also established.

scholarly journals BLOCK BOOTSTRAP CONSISTENCY UNDER WEAK ASSUMPTIONS

2018 ◽  
Vol 34 (6) ◽  
pp. 1383-1406 ◽  
Author(s):  
Gray Calhoun
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Partial Sums ◽  
Block Bootstrap ◽  
Mixing Processes ◽  
Bootstrap Methods ◽  
Moment Conditions ◽  
Sample Mean ◽  
Functional Clt

This paper weakens the size and moment conditions needed for typical block bootstrap methods (i.e., the moving blocks, circular blocks, and stationary bootstraps) to be valid for the sample mean of Near-Epoch-Dependent (NED) functions of mixing processes; they are consistent under the weakest conditions that ensure the original NED process obeys a central limit theorem (CLT), established by De Jong (1997, Econometric Theory 13(3), 353–367). In doing so, this paper extends De Jong’s method of proof, a blocking argument, to hold with random and unequal block lengths. This paper also proves that bootstrapped partial sums satisfy a functional CLT (FCLT) under the same conditions.

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