Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities. (with Bazant, M., Burch, D., and Lam, H.).

NOTE: This paper also circulated with the title “Beyond Edgeworth Expansions for Heavy-tailed Random Walks”

Summary:

Edgeworth expansions are corrections to the central limit theorem. They come in powers of n^(-[p-2]/2) for p integer at least 2 and are given in terms of Hermite polynomials. You can add a correction term of order n^(-[p-2]/2) to your central limit approximation if you have p moments. This paper answers the question: What happens if you run out of moments? Well, it turns out that you might need to add logarithmic factors in from of n^(-[p-2]/2) or NOT depending on the parity of the power-law tail behavior of your density. Also, different types of special functions (all of this is characterized in the paper) as the so-called Dawson integral.

Bibtex:

@Article{LamBlaBazBur10,

author = {H. Lam and J. Blanchet and M. Bazant and D. Burch},

title = {Corrections to the central limit theorem for heavy-tailed probability densities},

journal = {Journal of Theoretical Probability},

year = {Forthcoming},

volume = {},

pages = {}

}