scholarly journals Asymptotically Exact Constants in Natural Convergence Rate Estimates in the Lindeberg Theorem

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 501
Author(s):  
Ruslan Gabdullin ◽  
Vladimir Makarenko ◽  
Irina Shevtsova
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Convergence Rate ◽  
Rate Of Convergence ◽  
Central Limit ◽  
Lower Bounds ◽  
Third Order ◽  
Detailed Classification ◽  
Exact Constants

Following (Shevtsova, 2013) we introduce detailed classification of the asymptotically exact constants in natural estimates of the rate of convergence in the Lindeberg central limit theorem, namely in Esseen’s, Rozovskii’s, and Wang–Ahmad’s inequalities and their structural improvements obtained in our previous works. The above inequalities involve algebraic truncated third-order moments and the classical Lindeberg fraction and assume finiteness only the second-order moments of random summands. We present lower bounds for the introduced asymptotically exact constants as well as for the universal and for the most optimistic constants which turn to be not far from the upper ones.

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.

Characterizing the rate of convergence in the central limit theorem. II

1981 ◽  
Vol 89 (3) ◽  
pp. 511-523 ◽  
Author(s):  
Peter Hall
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Rate Of Convergence ◽  
Central Limit ◽  
Lower Bounds ◽  
Normal Approximation ◽  
Random Variables ◽  
Upper And Lower Bounds ◽  
The Central Limit Theorem ◽  
Order Of Magnitude

AbstractWe obtain upper and lower bounds of the same order of magnitude for the error between the distribution of a sum of independent and identically distributed random variables, and a normal approximation by a portion of a Chebychev-Cramér series. Our results are sufficiently general to contain the familiar characterizations by Ibragimov(4), Heyde and Leslie (3) and Lifshits(5), and complement some of those obtained earlier by the author (2).

Rate of convergence in the central limit theorem in the space D[0, 1]

1989 ◽  
Vol 28 (3) ◽  
pp. 229-238 ◽  
Author(s):  
V. Paulauskas ◽  
D. Jukneviciene
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Rate Of Convergence ◽  
Central Limit ◽  
The Central Limit Theorem
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