Journal of Theoretical Probability

, Volume 31, Issue 3, pp 1590–1605

# Approximate Central Limit Theorems

• Ben Berckmoes
• Geert Molenberghs
Article

## Abstract

We refine the classical Lindeberg–Feller central limit theorem by obtaining asymptotic bounds on the Kolmogorov distance, the Wasserstein distance, and the parameterized Prokhorov distances in terms of a Lindeberg index. We thus obtain more general approximate central limit theorems, which roughly state that the row-wise sums of a triangular array are approximately asymptotically normal if the array approximately satisfies Lindeberg’s condition. This allows us to continue to provide information in nonstandard settings in which the classical central limit theorem fails to hold. Stein’s method plays a key role in the development of this theory.

## Keywords

Kolmogorov metric Lindeberg–Feller central limit theorem Lindeberg index Prokhorov metric Stein’s method Wasserstein metric

60F05

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