Concentration inequalities for dependent random variables via the martingale method

Annals of Probability

Volumn 36, Issue 6, 2008, Pages 2126-2158

  Kontorovich, Leonid ^a   Ramanan, Kavita ^b  

^a WEIZMANN INSTITUTE OF SCIENCE   (Israel)

^b CARNEGIE MELLON UNIVERSITY   (United States)

Author keywords

Bounded martingale differences; Concentration inequality; Contracting markov chains; Hidden markov chains; Markov chains; McDiarmid's bound; Mixing coefficients

Indexed keywords

EID: 61849181973     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/07-AOP384     Document Type: Article

References (30)

1. AHLSWEDE, R., GÁGS, P. and KÖRNER, J. (1976). Bounds on conditional probabilities with applications in multi-user communication. Z. Wahrsch, Verw. Gebiete 34 157-177. MR0472221
2. AZUMA, K. (1967). Weighted sums of certain dependent random variables. Tôhoku Math. J. (2) 19 357-367. MR0221571
3. BEHRENDS. E. (2000). Introduction to Markov Chains. Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig. With special emphasis on rapid mixing. MR1730905
4. BOBKOV. S. G. and GÖTZE, F. (1999). Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 1-28. MR 1682772
5. BOBKOV, S. G. and LEDOUX, M. (1998). On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156 347-365. MR1636948
6. BOUCHERON, S., LUGOSI, G. and MASSART, P. (2003). Concentration inequalities using the entropy method. Ann. Probab. 31 1583-1614. MR 1989444
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. MR2278456
9. DEMBO, A. (1997). Information inequalities and concentration of measure. Ann. Probab. 25 927-939. MR1434131
10. DEMBO, A. and ZEITOUNI, O. (1996). Transportation approach to some concentration inequalities in product spaces. Electron. Comm. Probab. 1 83-90 (electronic). MR1423908
11. HOEFFDING, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13-30. MR0144363
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). MR 1399224
17. LEDOUX, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. American Mathematical Society, Providence, RI. MR 1849347
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. MR 1392329
20. MARTON, K. (1996). Bounding d̄-distance by informational divergence: A method to prove measure concentration. Ann. Probab. 24 857-866. MR1404531
21. MARTON, K. (1998). Measure concentration for a class of random processes. Probab. Theory Related Fields 110 427-439. MR1616492
22. MARTON, K. (2003). Measure concentration and strong mixing. Studia Sci. Math. Hungar. 40 95-113. MR2002993
23. MARTON, K. (2004). Measure concentration for Euclidean distance in the case of dependent random variables. Ann. Probab. 32 2526-2544. MR2078549
24. McDIARMID, C. (1989). On the method of bounded differences. In Surveys in Combinatorics, 1989 (Norwich, 1989). London Mathematical Society Lecture Note 25. Series 141 148-188. Cambridge Univ. Press, Cambridge. MR1036755
25. McDlARMID, C. (1998). Concentration. In Probabilistic Methods for Algorithmic Discrete Mathematics. Algorithms and Combinatorics 16 195-248. Springer, Berlin. MR 1678578
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. MR 1760620
27. SAMSON, P.-M. (2000). Concentration of measure inequalities for Markov chains and ∮- mixing processes. Ann. Probab. 28 416-461. MR 1756011
28. SCHECHTMAN, G. (2001). Concentration, Results and Applications. The Handbook in the Geometry of Banach Spaces. North-Holland, Amsterdam.
29. TALAGRAND, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81 73-205. MR1361756
30. TALAGRAND, M. (1996). New concentration inequalities in product spaces. Invent. Math. 126 505-563. MR1419006