Premium
On Bounds to the Rate of Convergence in the Central Limit Theorem
Author(s) -
Barbour A. D.,
Hall Peter
Publication year - 1985
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/17.2.151
Subject(s) - mathematics , independent and identically distributed random variables , random variable , central limit theorem , rate of convergence , combinatorics , zero (linguistics) , order (exchange) , moment (physics) , illustration of the central limit theorem , distribution (mathematics) , second moment of area , discrete mathematics , mathematical analysis , sum of normally distributed random variables , statistics , key (lock) , geometry , economics , ecology , linguistics , philosophy , physics , finance , classical mechanics , biology
Let( X j ) j ⩾ 1 be independent and identically distributed random variables with zero mean, unit variance and finite third absolute moment γ. The Berry‐Esséen theorem states thatsup x | p [ n − 1 2∑ j = 1 nX j ⩽ x] − φ ( x ) | ⩽ C γ n − 1 2and, because of the possibility of lattice valued variables X j , no improvement in the rate on the right hand side can be obtained, even though the rate, for many X j ‐distributions, is really much faster. In this paper, the departure of the distribution of a sum of independent but not necessarily identically distributed random variables from the normal is measured by a smoother metric than the uniform metric above, enabling a precise rate of convergence to be determined, within a tolerance which, in the above setting, is of order n ‐1 .