Concentration inequalities for dependent random variables via the martingale method

Concentration inequalities for dependent random variables via the martingale method

Leonid (Aryeh) Kontorovich and Kavita Ramanan

Source: Ann. Probab. Volume 36, Number 6 (2008), 2126-2158.

Abstract

The martingale method is used to establish concentration inequalities for a class of dependent random sequences on a countable state space, with the constants in the inequalities expressed in terms of certain mixing coefficients. Along the way, bounds are obtained on martingale differences associated with the random sequences, which may be of independent interest. As applications of the main result, concentration inequalities are also derived for inhomogeneous Markov chains and hidden Markov chains, and an extremal property associated with their martingale difference bounds is established. This work complements and generalizes certain concentration inequalities obtained by Marton and Samson, while also providing different proofs of some known results.

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Primary Subjects: 60E15

Secondary Subjects: 60J10, 60G42

Keywords: Concentration inequality; McDiarmid’s bound; bounded martingale differences; Markov chains; contracting Markov chains; hidden Markov chains; mixing coefficients

Full-text: Open access

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1229696598
Digital Object Identifier: doi:10.1214/07-AOP384
Mathematical Reviews number (MathSciNet): MR2478678
Zentralblatt MATH identifier: 1154.60310

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References

[1] Ahlswede, R., Gács, P. and Körner, J. (1976). Bounds on conditional probabilities with applications in multi-user communication. Z. Wahrsch. Verw. Gebiete 34 157–177.

[2] Azuma, K. (1967). Weighted sums of certain dependent random variables. Tôhoku Math. J. (2) 19 357–367.

Mathematical Reviews (MathSciNet): MR221571

Digital Object Identifier: doi:10.2748/tmj/1178243286

Project Euclid: euclid.tmj/1178243286

[3] Behrends, E. (2000). Introduction to Markov Chains. Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig. With special emphasis on rapid mixing.

Mathematical Reviews (MathSciNet): MR1730905

Zentralblatt MATH: 0953.60063

[4] Bobkov, S. G. and Götze, F. (1999). Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 1–28.

Mathematical Reviews (MathSciNet): MR1682772

Zentralblatt MATH: 0924.46027

Digital Object Identifier: doi:10.1006/jfan.1998.3326

[5] Bobkov, S. G. and Ledoux, M. (1998). On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156 347–365.

Mathematical Reviews (MathSciNet): MR1636948

Zentralblatt MATH: 0920.60002

Digital Object Identifier: doi:10.1006/jfan.1997.3187

[6] Boucheron, S., Lugosi, G. and Massart, P. (2003). Concentration inequalities using the entropy method. Ann. Probab. 31 1583–1614.

Mathematical Reviews (MathSciNet): MR1989444

Zentralblatt MATH: 1051.60020

Digital Object Identifier: doi:10.1214/aop/1055425791

Project Euclid: euclid.aop/1055425791

[7] Chatterjee, S. (2005). Concentration inequalities with exchangeable pairs. Ph.D. dissertation, Stanford Univ.

[8] Chazottes, J.-R., Collet, P., Külske, C. and Redig, F. (2007). Concentration inequalities for random fields via coupling. Probab. Theory Related Fields 137 201–225.

[9] Dembo, A. (1997). Information inequalities and concentration of measure. Ann. Probab. 25 927–939.

Mathematical Reviews (MathSciNet): MR1434131

Zentralblatt MATH: 0880.60018

Digital Object Identifier: doi:10.1214/aop/1024404424

Project Euclid: euclid.aop/1024404424

[10] Dembo, A. and Zeitouni, O. (1996). Transportation approach to some concentration inequalities in product spaces. Electron. Comm. Probab. 1 83–90 (electronic).

Mathematical Reviews (MathSciNet): MR1423908

Zentralblatt MATH: 0916.28003

[11] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.

Mathematical Reviews (MathSciNet): MR144363

Zentralblatt MATH: 0127.10602

Digital Object Identifier: doi:10.2307/2282952

JSTOR: links.jstor.org

[12] Kontoyiannis, I., Lastras-Montano, L. A. and Meyn, S. (2005). Relative entropy and exponential deviation bounds for general Markov chains. In Proc. of Int. Symp. Inf. Theor. (ISIT), Adelaide, Australia.

[13] Kontorovich, L. (2006). Measure concentration of hidden Markov processes. Available at http://arxiv.org/abs/math.PR/0608064.

[14] Kontorovich, L. (2006). Measure concentration of Markov tree processes. Available at http://arxiv.org/abs/math.PR/0608511.

[15] Kontorovich, L. (2007). Measure concentration of strongly mixing processes with applications Ph.D. dissertation, Carnegie Mellon Univ.

[16] Ledoux, M. (1995/97). On Talagrand’s deviation inequalities for product measures. ESAIM Probab. Statist. 1 63–87 (electronic).

Mathematical Reviews (MathSciNet): MR1399224

Digital Object Identifier: doi:10.1051/ps:1997103

[17] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. American Mathematical Society, Providence, RI.

Mathematical Reviews (MathSciNet): MR1849347

[18] Markov, A. A. (1906). Extension of the law of large numbers to dependent quantities. Izvestiia Fiz.-Matem. Obsch. Kazan Univ. 15 135–156.

[19] Marton, K. (1995). A measure concentration inequality for contracting Markov chains. Geom. Funct. Anal. 6 556–571.

Mathematical Reviews (MathSciNet): MR1392329

Digital Object Identifier: doi:10.1007/BF02249263

[20] Marton, K. (1996). Bounding d̅-distance by informational divergence: A method to prove measure concentration. Ann. Probab. 24 857–866.

Mathematical Reviews (MathSciNet): MR1404531

Zentralblatt MATH: 0865.60017

Digital Object Identifier: doi:10.1214/aop/1039639365

Project Euclid: euclid.aop/1039639365

[21] Marton, K. (1998). Measure concentration for a class of random processes. Probab. Theory Related Fields 110 427–439.

Mathematical Reviews (MathSciNet): MR1616492

Zentralblatt MATH: 0927.60050

Digital Object Identifier: doi:10.1007/s004400050154

[22] Marton, K. (2003). Measure concentration and strong mixing. Studia Sci. Math. Hungar. 40 95–113.

Mathematical Reviews (MathSciNet): MR2002993

Digital Object Identifier: doi:10.1556/SScMath.40.2003.1-2.8

[23] Marton, K. (2004). Measure concentration for Euclidean distance in the case of dependent random variables. Ann. Probab. 32 2526–2544.

Mathematical Reviews (MathSciNet): MR2078549

Digital Object Identifier: doi:10.1214/009117904000000702

Project Euclid: euclid.aop/1091813622

[24] McDiarmid, C. (1989). On the method of bounded differences. In Surveys in Combinatorics, 1989 (Norwich, 1989). London Math. Soc. Lecture Note Ser. 141 148–188. Cambridge Univ. Press, Cambridge.

Mathematical Reviews (MathSciNet): MR1036755

Zentralblatt MATH: 0712.05012

[25] McDiarmid, C. (1998). Concentration. In Probabilistic Methods for Algorithmic Discrete Mathematics. Algorithms Combin. 16 195–248. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR1678578

[26] Otto, F. and Villani, C. (2000). Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 361–400.

Mathematical Reviews (MathSciNet): MR1760620

Zentralblatt MATH: 0985.58019

Digital Object Identifier: doi:10.1006/jfan.1999.3557

[27] Samson, P.-M. (2000). Concentration of measure inequalities for Markov chains and Φ-mixing processes. Ann. Probab. 28 416–461.

Mathematical Reviews (MathSciNet): MR1756011

Zentralblatt MATH: 1044.60061

Digital Object Identifier: doi:10.1214/aop/1019160125

Project Euclid: euclid.aop/1019160125

[28] Schechtman, G. (2001). Concentration, Results and Applications. The Handbook in the Geometry of Banach Spaces. North-Holland, Amsterdam.

Mathematical Reviews (MathSciNet): MR1999604

Zentralblatt MATH: 1057.46011

Digital Object Identifier: doi:10.1016/S1874-5849(03)80044-X

[29] Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81 73–205.

Mathematical Reviews (MathSciNet): MR1361756

Digital Object Identifier: doi:10.1007/BF02699376

[30] Talagrand, M. (1996). New concentration inequalities in product spaces. Invent. Math. 126 505–563.

Mathematical Reviews (MathSciNet): MR1419006

Zentralblatt MATH: 0893.60001

Digital Object Identifier: doi:10.1007/s002220050108

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