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STEIN COUPLINGS FOR BERRY-ESSEEN BOUNDS AND
CONCENTRATION INEQUALITIES
by
Subhankar Ghosh
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Ful llment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(APPLIED MATHEMATICS)
May 2012
Copyright 2012 Subhankar Ghosh
Object Description
| Title | Stein couplings for Berry-Esseen bounds and concentration inequalities |
| Author | Ghosh, Subhankar |
| Author email | subhankar.ghosh@gmail.com;subhankar.ghosh@sas.com |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Applied Mathematics |
| School | College of Letters, Arts And Sciences |
| Date defended/completed | 2012-01-13 |
| Date submitted | 2012-05-07 |
| Date approved | 2012-05-08 |
| Restricted until | 2012-05-08 |
| Date published | 2012-05-08 |
| Advisor (committee chair) | Goldstein, Larry |
| Advisor (committee member) |
Alexander, Kenneth S. Langholz, Brian |
| Abstract | Stein's method is one of the cornerstones of modern limit theory in probability. While working on the problem of inadmissibility of the multivariate sample mean for Gaussian distribution, Charles Stein used a characterization identity of the normal distribution. In 1972, he showed that the same identity could be used to prove the Central Limit Theorem in cases where summands were not even independent and also to yield the rate of convergence to normal distribution. For probability theorists, this new method was appealing as it did not use Fourier transforms. Within a few years many new CLT error bounds were obtained for cases where the summands were not independent. Louis Chen showed that similar ideas could be used for obtaining error bounds in the realm of Poisson approximation also. From the early days of Stein's method, coupled random variables played an important role in Stein's method and some of the very useful coupling techniques were devised in the late eighties and nineties. In this thesis, we will look at two of these couplings namely the zero and size bias couplings. In particular, we will show how zero bias couplings can be used to obtain error bounds in a combinatorial central limit theorem using involutions. We furthermore show that the couplings are useful not only for obtaining error bounds but for obtaining concentration of measure inequalities as well. We illustrate the use of our results through several nontrivial examples. |
| Keyword | probability; mathematics; Stein's method |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Provenance | Electronically uploaded by the author |
| Type | texts |
| Legacy record ID | usctheses-m |
| Rights | Ghosh, Subhankar |
| Access conditions | The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright. The original signature page accompanying the original submission of the work to the USC Libraries is retained by the USC Libraries and a copy of it may be obtained by authorized requesters contacting the repository e-mail address given. |
| Repository name | University of Southern California Digital Library |
| Repository address | USC Digital Library, University of Southern California, University Park Campus MC 7002, 106 University Village, Los Angeles, California 90089-7002, USA |
| Repository email | cisadmin@usc.edu |
| Archival file | uscthesesreloadpub_Volume4/etd-GhoshSubha-815.pdf |
Description
| Title | Page 1 |
| Full text | STEIN COUPLINGS FOR BERRY-ESSEEN BOUNDS AND CONCENTRATION INEQUALITIES by Subhankar Ghosh A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) May 2012 Copyright 2012 Subhankar Ghosh |
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