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A new approach to strong embeddings
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  • Published: 08 October 2010

A new approach to strong embeddings

  • Sourav Chatterjee1 

Probability Theory and Related Fields volume 152, pages 231–264 (2012)Cite this article

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Abstract

We revisit strong approximation theory from a new perspective, culminating in a proof of the Komlós–Major–Tusnády embedding theorem for the simple random walk. The proof is almost entirely based on a series of soft arguments and easy inequalities. The new technique, inspired by Stein’s method of normal approximation, is applicable to any setting where Stein’s method works. In particular, one can hope to take it beyond sums of independent random variables.

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Acknowledgments

The author is particularly indebted to David Mason and Andrei Zaitsev for clearing up many misconceptions about the literature and providing very helpful guidance. The author thanks Miklós Csörgő, Persi Diaconis, Yuval Peres, Peter Bickel, Craig Evans, and Raghu Varadhan for useful discussions and advice; and Ron Peled, Arnab Sen, Partha Dey, and Shankar Bhamidi for comments about the manuscript. Special thanks are due to Partha Dey for a careful verification of the proofs.

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This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Sourav Chatterjee

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  1. Sourav Chatterjee
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Corresponding author

Correspondence to Sourav Chatterjee.

Additional information

S. Chatterjee’s research was partially supported by NSF grant DMS-0707054 and a Sloan Research Fellowship.

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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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Cite this article

Chatterjee, S. A new approach to strong embeddings. Probab. Theory Relat. Fields 152, 231–264 (2012). https://doi.org/10.1007/s00440-010-0321-8

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  • Received: 21 July 2008

  • Revised: 02 July 2010

  • Published: 08 October 2010

  • Issue Date: February 2012

  • DOI: https://doi.org/10.1007/s00440-010-0321-8

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Keywords

  • Strong embedding
  • KMT embedding
  • Stein’s method

Mathematics Subject Classification (2000)

  • 60F17
  • 60F99
  • 60G50
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