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Classical Edgeworth expansions provide asymptotic correction terms to theCentral Limit Theorem (CLT) up to an order that depends on the number ofmoments available. In this paper, we provide subsequent correction terms beyondthose given by a standard Edgeworth expansion in the general case of regularlyvarying distributions with diverging moments (beyond the second). Thesubsequent terms can be expressed in a simple closed form in terms of certainspecial functions (Dawson's integral and parabolic cylinder functions), andthere are qualitative differences depending on whether the number of momentsavailable is even, odd or not an integer, and whether the distributions aresymmetric or not. If the increments have an even number of moments, thenadditional logarithmic corrections must also be incorporated in the expansionparameter. An interesting feature of our correction terms for the CLT is thatthey become dominant outside the central region and blend naturally with knownlarge-deviation asymptotics when these are applied formally to the spatialscales of the CLT.

Corrections to the Central Limit Theorem for Heavy-Tailed Probability Densities