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 regu...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | en_US |
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Springer-Verlag
2013
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| Online Access: | http://hdl.handle.net/1721.1/77920 |
| _version_ | 1811090479060615168 |
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| author | Lam, Henry Blanchet, Jose Burch, Damian Bazant, Martin Z. |
| author2 | Massachusetts Institute of Technology. Department of Chemical Engineering |
| author_facet | Massachusetts Institute of Technology. Department of Chemical Engineering Lam, Henry Blanchet, Jose Burch, Damian Bazant, Martin Z. |
| author_sort | Lam, Henry |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T14:46:48Z |
| format | Article |
| id | mit-1721.1/77920 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T14:46:48Z |
| publishDate | 2013 |
| publisher | Springer-Verlag |
| record_format | dspace |
| spelling | mit-1721.1/779202022-09-29T10:29:33Z Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities Lam, Henry Blanchet, Jose Burch, Damian Bazant, Martin Z. Massachusetts Institute of Technology. Department of Chemical Engineering Massachusetts Institute of Technology. Department of Mathematics Burch, Damian Bazant, Martin Z. 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. 2013-03-15T18:40:38Z 2013-03-15T18:40:38Z 2011-09 2011-04 Article http://purl.org/eprint/type/JournalArticle 0894-9840 1572-9230 http://hdl.handle.net/1721.1/77920 Lam, Henry et al. “Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities.” Journal of Theoretical Probability 24.4 (2011): 895–927. en_US http://dx.doi.org/10.1007/s10959-011-0379-y Journal of Theoretical Probability Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf Springer-Verlag Other university web domain |
| spellingShingle | Lam, Henry Blanchet, Jose Burch, Damian Bazant, Martin Z. Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities |
| title | Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities |
| title_full | Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities |
| title_fullStr | Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities |
| title_full_unstemmed | Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities |
| title_short | Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities |
| title_sort | corrections to the central limit theorem for heavy tailed probability densities |
| url | http://hdl.handle.net/1721.1/77920 |
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