A Leading Term Approach to Asymptotic Expansions in the Central Limit Theorem

1984 ◽  
Vol s3-49 (3) ◽  
pp. 423-444 ◽  
Author(s):  
Peter Hall
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Asymptotic Expansions ◽  
The Central Limit Theorem ◽  
Leading Term

On the fluctuations of eigenvalues of multiplicative deformed unitary invariant ensembles

2016 ◽  
Vol 05 (02) ◽  
pp. 1650007 ◽  
Author(s):  
Vladimir Vasilchuk
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Matrices ◽  
Central Limit ◽  
Counting Measure ◽  
Linear Eigenvalue Statistics ◽  
The Central Limit Theorem ◽  
Eigenvalue Statistics ◽  
Leading Term

We consider the ensemble of [Formula: see text] random matrices [Formula: see text], where [Formula: see text] and [Formula: see text] are non-random, unitary, having the limiting Normalized Counting Measure (NCM) of eigenvalues, and [Formula: see text] is unitary, uniformly distributed over [Formula: see text]. We find the leading term of the covariance of traces of resolvent of [Formula: see text] and establish the Central Limit Theorem for sufficiently smooth linear eigenvalue statistics of [Formula: see text] as [Formula: see text].

scholarly journals On non-uniform and global descriptions of the rate of convergence of Asymptotic expansions in the central limit theorem

1986 ◽  
Vol 41 (3) ◽  
pp. 326-335 ◽  
Author(s):  
Peter Hall ◽  
T. Nakata
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Rate Of Convergence ◽  
Central Limit ◽  
Lower Bounds ◽  
Rates Of Convergence ◽  
Upper And Lower Bounds ◽  
The Central Limit Theorem ◽  
Order Of Magnitude ◽  
Leading Term

AbstractThe leading term approach to rates of convergence is employed to derive non-uniform and global descriptions of the rate of convergence in the central limit theorem. Both upper and lower bounds are obtained, being of the same order of magnitude, modulo terms of order n-r. We are able to derive general results by considering only those expansions with an odd number of terms.

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