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May 2018 Chaining, interpolation and convexity II: The contraction principle
Ramon van Handel
Ann. Probab. 46(3): 1764-1805 (May 2018). DOI: 10.1214/17-AOP1214

Abstract

The generic chaining method provides a sharp description of the suprema of many random processes in terms of the geometry of their index sets. The chaining functionals that arise in this theory are however notoriously difficult to control in any given situation. In the first paper in this series, we introduced a particularly simple method for producing the requisite multiscale geometry by means of real interpolation. This method is easy to use, but does not always yield sharp bounds on chaining functionals. In the present paper, we show that a refinement of the interpolation method provides a canonical mechanism for controlling chaining functionals. The key innovation is a simple but powerful contraction principle that makes it possible to efficiently exploit interpolation. We illustrate the utility of this approach by developing new dimension-free bounds on the norms of random matrices and on chaining functionals in Banach lattices. As another application, we give a remarkably short interpolation proof of the majorizing measure theorem that entirely avoids the greedy construction that lies at the heart of earlier proofs.

Citation

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Ramon van Handel. "Chaining, interpolation and convexity II: The contraction principle." Ann. Probab. 46 (3) 1764 - 1805, May 2018. https://doi.org/10.1214/17-AOP1214

Information

Received: 1 November 2016; Revised: 1 July 2017; Published: May 2018
First available in Project Euclid: 12 April 2018

zbMATH: 06894785
MathSciNet: MR3785599
Digital Object Identifier: 10.1214/17-AOP1214

Subjects:
Primary: 41A46, 46B20, 46B70, 60B11, 60G15

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 3 • May 2018
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