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de Acosta, A.; Ney, Peter

Large deviation lower bounds for arbitrary additive functionals of a Markov chain. (English) Zbl 0936.60022

Ann. Probab. 26, No. 4, 1660-1682 (1998).
Summary: A universal large deviation lower bound is proved for sums of Banach space valued functions of an irreducible, general state space Markov chain. There are no restrictions on the functions (other than measurability).

Cited in 9 Documents

MSC:

60F10 Large deviations
60J55 Local time and additive functionals

Keywords:

large deviations; Markov chains; convergence parameter; Banach space
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References:

[1] ATHREYA, K. B. and NEY, P. E. 1978. A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc. 245 493 501. · Zbl 0397.60053 · doi:10.2307/1998882
[2] AZENCOTT, R. 1980. Grandes deviations et applications. Lecture Notes in Math. 774 1 176. Springer, Berlin. · Zbl 0435.60028
[3] BAHADUR, R. R. and ZABELL, S. L. 1979. Large deviations of the sample mean in general vector spaces. Ann. Probab. 7 587 621. · Zbl 0424.60028 · doi:10.1214/aop/1176994985
[4] BOLTHAUSEN, E. 1987. Markov process large deviation in the -topology. Stochastic Process. Appl. 33 1 27. · Zbl 0625.60026 · doi:10.1016/0304-4149(87)90192-X
[5] BRYC, W. and DEMBO, A. 1996. Large deviations and strong mixing. Ann. Inst. H. Poincare 32 549 569. \' · Zbl 0863.60028
[6] CONWAY, J. B. 1985. A Course in Functional Analysis. Springer, Berlin. · Zbl 0558.46001
[7] DEMBO, A. and ZEITOUNI, O. 1993. Large Deviations Techniques and Applications. Jones and Bartlett, Boston. · Zbl 0793.60030
[8] DINWOODIE, I. H. 1993. Identifying a large deviation rate function. Ann. Probab. 21 216 231. · Zbl 0777.60024 · doi:10.1214/aop/1176989402
[9] DINWOODIE, I. H. and NEY, P. E. 1995. Occupation measures for Markov chains. J. Theoret. Probab. 8 679 691. · Zbl 0834.60029 · doi:10.1007/BF02218050
[10] DONSKER, M. D. and VARADHAN, S. R. S. 1975, 1976. Asymptotic evaluation of certain Markov process expectations for large time. I, II, III. Comm. Pure Appl. Math. 28 1 47; 28 279 301; 29 389 461. · Zbl 0348.60032 · doi:10.1002/cpa.3160290405
[11] EKELAND, I. and TEMAM, R. 1976. Convex Analysis and Variational Problems. NorthHolland, Amsterdam. · Zbl 0322.90046
[12] JAIN, N. C. 1990. Large deviation lower bounds for additive functionals of Markov processes. Ann. Probab. 18 1071 1098. · Zbl 0713.60037 · doi:10.1214/aop/1176990736
[13] LANFORD, O. E. 1973. Entropy and equilibrium states in classical statistical mechanics. Lecture Notes in Phys. 20 1 113. Springer, Berlin.
[14] MEYN, S. P. and TWEEDIE, R. L. 1993. Markov Chains and Stochastic Stability. Springer, London. · Zbl 0925.60001
[15] MILLER, H. 1961. A convexity property in the theory of random variables defined on a finite Markov chain. Ann. Math. Statist. 32 1260 1270. · Zbl 0108.15101 · doi:10.1214/aoms/1177704865
[16] NEY, P. E. and NUMMELIN, E. 1987. Markov additive processes. I, II. Ann. Probab. 15 561 592; 593 609. · Zbl 0625.60027 · doi:10.1214/aop/1176992159
[17] NUMMELIN, E. 1984. General Irreducible Markov Chains and Non-Negative Operators. Cambridge Univ. Press. · Zbl 0551.60066
[18] SHURENKOV, V. M. 1992. On the relationship between spectral radii and Perron roots. Technical report, Dept. Mathematics Goteborg Univ. \"
[19] STROOCK, D. W. 1984. An Introduction to the Theory of Large Deviations. Springer, Berlin. · Zbl 0552.60022
[20] VARADHAN, S. R. S. 1984. Large Deviations and Applications. SIAM, Philadelphia. · Zbl 0549.60023
[21] CLEVELAND, OHIO 44106 MADISON, WISCONSIN 53706 E-MAIL: ney@math.wisc.edu
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