The Berry-Esseen Theorem sharpens the central limit theorem by giving an explicit bound on how far the distribution of a normalized sum of independent random variables can be from the normal distribution for any finite sample size, with the bound shrinking in proportion to one over the square root of the number of variables summed. Named for Andrew C. Berry and Carl-Gustav Esseen, it quantifies the rate of convergence that the classical central limit theorem states only in the limit.
Facts
StatementFor a sum of independent, identically distributed random variables with finite third absolute moment, the theorem bounds how far the exact distribution of the standardized sum can be from the standard normal distribution by a constant multiple of the third absolute moment divided by the cube of the standard deviation and the square root of the sample size. Andrew C. Berry proved an early version in 1941 and Carl-Gustav Esseen an independent, sharper version in 1942. 1 Sources
1. Berry-Esseen Theorem (Wikipedia)
Wikimedia FoundationStatement of the theorem section, first sentenceQuote, Statement of the theorem section, first sentence
Statements of the theorem vary, as it was independently discovered by two mathematicians, Andrew C. Berry (in 1941) and Carl-Gustav Esseen (1942), who then, along with other authors, refined it repeatedly over subsequent decades.
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