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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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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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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DOI: https://doi.org/10.1007/s00440-010-0321-8
Keywords
- Strong embedding
- KMT embedding
- Stein’s method
Mathematics Subject Classification (2000)
- 60F17
- 60F99
- 60G50