4 research outputs found
Corrections to the Central Limit Theorem for Heavy-Tailed Probability Densities
Classical Edgeworth expansions provide asymptotic correction terms to the
Central Limit Theorem (CLT) up to an order that depends on the number of
moments available. In this paper, we provide subsequent correction terms beyond
those given by a standard Edgeworth expansion in the general case of regularly
varying distributions with diverging moments (beyond the second). The
subsequent terms can be expressed in a simple closed form in terms of certain
special functions (Dawson's integral and parabolic cylinder functions), and
there are qualitative differences depending on whether the number of moments
available is even, odd or not an integer, and whether the distributions are
symmetric or not. If the increments have an even number of moments, then
additional logarithmic corrections must also be incorporated in the expansion
parameter. An interesting feature of our correction terms for the CLT is that
they become dominant outside the central region and blend naturally with known
large-deviation asymptotics when these are applied formally to the spatial
scales of the CLT