Electronic Journal of Probability

The convex distance inequality for dependent random variables, with applications to the stochastic travelling salesman and other problems

Daniel Paulin

Full-text: Open access

Abstract

We prove concentration inequalities for general functions of weakly dependent random variables satisfying the Dobrushin condition. In particular, we show Talagrand's convex distance inequality for this type of dependence. We apply our bounds to a version of the stochastic salesman problem, the Steiner tree problem, the total magnetisation of the Curie-Weiss model with external field, and exponential random graph models. Our proof uses the exchangeable pair method for proving concentration inequalities introduced by Chatterjee (2005). Another key ingredient of the proof is a subclass of $(a,b)$-self-bounding functions, introduced by Boucheron, Lugosi and Massart (2009).

Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 68, 34 pp.

Dates
Accepted: 11 August 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065710

Digital Object Identifier
doi:10.1214/EJP.v19-3261

Mathematical Reviews number (MathSciNet)
MR3248197

Zentralblatt MATH identifier
1330.60039

Subjects
Primary: 60E15: Inequalities; stochastic orderings
Secondary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
concentration inequalities Stein's method exchangeable pairs reversible Markov chains stochastic travelling salesman problem Steiner tree sampling without replacement Dobrushin condition exponential random graph

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Paulin, Daniel. The convex distance inequality for dependent random variables, with applications to the stochastic travelling salesman and other problems. Electron. J. Probab. 19 (2014), paper no. 68, 34 pp. doi:10.1214/EJP.v19-3261. https://projecteuclid.org/euclid.ejp/1465065710


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