On Bounds to the Rate of Convergence in the Central Limit Theorem

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 .