# Chernoff and Berry–Esséen inequalities for Markov processes

ESAIM: Probability and Statistics (2001)

- Volume: 5, page 183-201
- ISSN: 1292-8100

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topLezaud, Pascal. "Chernoff and Berry–Esséen inequalities for Markov processes." ESAIM: Probability and Statistics 5 (2001): 183-201. <http://eudml.org/doc/104272>.

@article{Lezaud2001,

abstract = {In this paper, we develop bounds on the distribution function of the empirical mean for general ergodic Markov processes having a spectral gap. Our approach is based on the perturbation theory for linear operators, following the technique introduced by Gillman.},

author = {Lezaud, Pascal},

journal = {ESAIM: Probability and Statistics},

keywords = {Markov process; Chernoff bound; Berry–Esséen; eigenvalues; perturbation theory; Berry-Esséen},

language = {eng},

pages = {183-201},

publisher = {EDP-Sciences},

title = {Chernoff and Berry–Esséen inequalities for Markov processes},

url = {http://eudml.org/doc/104272},

volume = {5},

year = {2001},

}

TY - JOUR

AU - Lezaud, Pascal

TI - Chernoff and Berry–Esséen inequalities for Markov processes

JO - ESAIM: Probability and Statistics

PY - 2001

PB - EDP-Sciences

VL - 5

SP - 183

EP - 201

AB - In this paper, we develop bounds on the distribution function of the empirical mean for general ergodic Markov processes having a spectral gap. Our approach is based on the perturbation theory for linear operators, following the technique introduced by Gillman.

LA - eng

KW - Markov process; Chernoff bound; Berry–Esséen; eigenvalues; perturbation theory; Berry-Esséen

UR - http://eudml.org/doc/104272

ER -

## References

top- [1] D. Aldous and J. Fill, Reversible Markov Chains and Random Walks on Graphs. Monograph in preparation. Available from the Aldous’s home page at http://www.stat.berkeley.edu/users/aldous/book.html
- [2] B. Bercu and A. Rouault, Sharp large deviations for the Ornstein–Uhlenbeck process (to appear). Zbl1101.60320
- [3] E. Bolthausen, The Berry–Esseen Theorem for Functionals of Discrete Markov Chains. Z. Wahrscheinlichkeitstheorie Verw. 54 (1980) 59-73. Zbl0431.60019
- [4] W. Bryc and A. Dembo, Large deviations for quadratic functionals of gaussian processes. J. Theoret. Probab. 10 (1997) 307-332. Zbl0894.60026MR1455147
- [5] M.F. Cheng and F.Y. Wang, Estimation of spectral gap for elliptic operators. Trans. AMS 349 (1997) 1239-1267. Zbl0872.35072MR1401516
- [6] K.L. Chung. Markov chains with stationnary transition probabilities. Springer-Verlag (1960). Zbl0092.34304MR116388
- [7] J.D. Deuschel and D.W. Stroock, Large Deviations. Academic Press, Boston (1989). Zbl0705.60029MR997938
- [8] P. Diaconis, S. Holmes and R.M. Neal, Analysis of a non-reversible markov chain sampler, Technical Report. Cornell University, BU-1385-M, Biometrics Unit (1997). Zbl1083.60516
- [9] I.H. Dinwoodie, A probability inequality for the occupation measure of a reversible Markov chain. Ann. Appl. Probab 5 (1995) 37-43. Zbl0829.60022MR1325039
- [10] I.H. Dinwoodie, Expectations for nonreversible Markov chains. J. Math. Ann. App. 220 (1998) 585-596. Zbl0946.60067MR1614916
- [11] I.H. Dinwoodie and P Ney, Occupation measures for Markov chains. J. Theoret. Probab. 8 (1995) 679-691. Zbl0834.60029MR1340833
- [12] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2. Wiley & Sons, 2nd Edition (1971). Zbl0219.60003MR270403
- [13] S. Gallot and D. Hulin and J. Lafontaine, Riemannian Geometry. Springer-Verlag (1990). Zbl0716.53001MR1083149
- [14] D. Gillman, Hidden Markov Chains: Rates of Convergence and the Complexity of Inference, Ph.D. Thesis. Massachusetts Institute of Technology (1993).
- [15] L. Gross, Logarithmic Sobolev Inequalities and Contractivity Properties of Semigroups, in Dirichlet forms, Varenna (Italy). Springer-Verlag, Lecture Notes in Math. 1563 (1992) 54-88. Zbl0812.47037MR1292277
- [16] J.L. Jensen, Saddlepoint Approximations. Oxford Statist. Sci. Ser. 16.
- [17] T. Kato, Perturbation theory for linear operators. Springer (1966). Zbl0148.12601
- [18] D. Landers and L. Rogge, On the rate of convergence in the central limit theorem for Markov chains. Z. Wahrscheinlichkeitstheorie Verw. 35 (1976) 169-183. Zbl0315.60014MR407938
- [19] G.F. Lawler and A.D. Sokal, Bounds on the ${L}^{2}$ spectrum for Markov chains and Markov processes: A generalization of Cheeger’s inequality. Trans. Amer. Math. Soc. 309 (1988) 557-580. Zbl0716.60073
- [20] P. Lezaud, Chernoff-type Bound for Finite Markov Chains. Ann. Appl. Probab 8 (1998) 849-867. Zbl0938.60027MR1627795
- [21] B. Mann, Berry–Esseen Central Limit Theorem for Markov chains, Ph.D. Thesis. Harvard University (1996).
- [22] K. Marton, A measure concentration inequality for contracting Markov chains. Geom. Funct. Anal. 6 (1996) 556-571. Zbl0856.60072MR1392329
- [23] S.V. Nagaev, Some limit theorems for stationary Markov chains. Theory Probab. Appl. 2 (1957) 378-406. Zbl0078.31804MR94846
- [24] P.M. Samson, Concentration of measure inequalities for Markov chains and $\phi $-mixing processes, Ann. Probab. 28 (2000) 416-461. Zbl1044.60061MR1756011
- [25] H.F. Trotter, On the product of semi-groups of operators. Proc. Amer. Math. Soc. 10 (1959) 545-551. Zbl0099.10401MR108732
- [26] F.Y. Wang, Existence of spectral gap for elliptic operators. Math. Sci. Res. Inst. (1998). MR1714760

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