A large deviation inequality for vector functions on finite reversible Markov chains

Annals of Applied Probability

Volumn 17, Issue 4, 2007, Pages 1202-1221

  Kargin, Vladislav ^a,b  

^a NEW YORK UNIVERSITY   (United States)

^b Institute of Mathematical Sciences   (United States)

Author keywords

Bernstein inequality; Hoeffding inequality; Large deviations; Markov chain; Spectral gap

Indexed keywords

EID: 39449109688     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/105051607000000078     Document Type: Article

References (20)

1. ALDOUS, D. and FILL, J. (2006). Reversible Markov chains and random walks on graphs. Monograph in preparation. Available at http://128.32.135.2/users/aldous/RWG/book. html.
2. BENNETT, G. (1962). Probability inequalities for the sum of independent random variables. J. Amer. Statist. Assoc. 57 33-45.
3. BERNSTEIN, S. (1952). Collected Papers. Izdat. Acad. Nauk SSSR, Moscow. (In Russian.) MR0048360
4. CHERNOFF, H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Statist. 23 493-507. MR0057518
5. DIACONIS, P. (1988). Group Representations in Probability and Statistics. IMS, Hayward, CA. MR0964069
6. DINWOODIE, I. H. (1995). A probability inequality for the occupation measure of a reversible Markov chain. Ann. Appl. Probab. 5 37-43. MR1325039
7. DONSKER, M. D. and VARADHAN, S. R. S. (1975). Asymptotic evaluation of certain Markov process expectations for large time. I, II. Comm. Pure Appl. Math. 28 1-47, 279-301. MR0386024
8. GANGOLLI, A. R. (1991). Convergence bounds for Markov chains and applications to sampling. Ph.D. dissertation, Stanford Univ.
9. GILLMAN, D. (1993). Hidden Markov chains: Convergence rates and the complexity of inference. Ph.D. thesis, MIT.
10. HOEFFDING, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13-30. MR0144363
11. KALLENBERG, O. and SZTENCEL, R. (1991). Some dimension-free features of vector-valued martingales. Probab. Theory Related Fields 88 215-247. MR1096481
12. KATO, T. (1980). Perturbation Theory for Linear Operators. Springer, Berlin. MR0407617
13. KOLMOGOROFF, A. (1929). Über das Gesetz des iterierten Logarithmus. Math. Ann. 101 126-135. MR1512520
14. LEZAUD, P. (1998). Chernoff-type bound for finite Markov chains. Ann. Appl. Probab. 8 849-867. MR1627795
15. MILLER, H. D. (1961). A convexity property in the theory of random variables defined on a finite Markov chain. Ann. Math. Statist. 32 1260-1270. MR0126886
16. NAGAEV, S. V. (1957). Some limit theorems for stationary Markov chains. Theory Probab. Appl. 2 378-406. MR0094846
17. PROKHOROV, Y. V. (1959). An extremal problem in probability theory. Theory Probab. Appl. 4 201-203. MR0121857
18. PROKHOROV, Y. V. (1968). An extension of S. N. Bernstein's inequalities to multidimensional distributions. Theory Probab. Appl. 13 260-267. MR0230353
19. SALOFF-COSTE, L. (2004). Random walks on finite groups. In Probability on Discrete Structures. Encyclopaedia of Mathematical Sciences (H. Kesten, ed.) 100 263-346. Springer, Berlin. MR2023654
20. YURINSKII, V. V. (1970). On an infinite-dimensional version of S. N. Bernstein's inequalities. Theory Probab. Appl. 15 108-109. MR0268941