The Annals of Probability

Large deviation lower bounds for arbitrary additive functionals of a Markov chain

Peter Ney and A. de Acosta

Full-text: Open access

Abstract

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).

Article information

Source
Ann. Probab., Volume 26, Number 4 (1998), 1660-1682.

Dates
First available in Project Euclid: 31 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022855877

Digital Object Identifier
doi:10.1214/aop/1022855877

Mathematical Reviews number (MathSciNet)
MR1675055

Zentralblatt MATH identifier
0936.60022

Subjects
Primary: 60F10: Large deviations 60J55: Local time and additive functionals

Keywords
Large deviations Markov chains convergence parameter Banach space

Citation

de Acosta, A.; Ney, Peter. Large deviation lower bounds for arbitrary additive functionals of a Markov chain. Ann. Probab. 26 (1998), no. 4, 1660--1682. doi:10.1214/aop/1022855877. https://projecteuclid.org/euclid.aop/1022855877


Export citation

References

  • 5 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.
  • 6 AZENCOTT, R. 1980. Grandes deviations et applications. Lecture Notes in Math. 774 1 176. Springer, Berlin.
  • 7 BAHADUR, R. R. and ZABELL, S. L. 1979. Large deviations of the sample mean in general vector spaces. Ann. Probab. 7 587 621.
  • 8 BOLTHAUSEN, E. 1987. Markov process large deviation in the -topology. Stochastic Process. Appl. 33 1 27.
  • 9 BRYC, W. and DEMBO, A. 1996. Large deviations and strong mixing. Ann. Inst. H. Poincare 32 549 569. ´
  • 10 CONWAY, J. B. 1985. A Course in Functional Analysis. Springer, Berlin.
  • 11 DEMBO, A. and ZEITOUNI, O. 1993. Large Deviations Techniques and Applications. Jones and Bartlett, Boston.
  • 12 DINWOODIE, I. H. 1993. Identifying a large deviation rate function. Ann. Probab. 21 216 231.
  • 13 DINWOODIE, I. H. and NEY, P. E. 1995. Occupation measures for Markov chains. J. Theoret. Probab. 8 679 691.
  • 14 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.
  • 15 EKELAND, I. and TEMAM, R. 1976. Convex Analysis and Variational Problems. NorthHolland, Amsterdam.
  • 16 JAIN, N. C. 1990. Large deviation lower bounds for additive functionals of Markov processes. Ann. Probab. 18 1071 1098.
  • 17 LANFORD, O. E. 1973. Entropy and equilibrium states in classical statistical mechanics. Lecture Notes in Phys. 20 1 113. Springer, Berlin.
  • 18 MEYN, S. P. and TWEEDIE, R. L. 1993. Markov Chains and Stochastic Stability. Springer, London.
  • 19 MILLER, H. 1961. A convexity property in the theory of random variables defined on a finite Markov chain. Ann. Math. Statist. 32 1260 1270.
  • 20 NEY, P. E. and NUMMELIN, E. 1987. Markov additive processes. I, II. Ann. Probab. 15 561 592; 593 609.
  • 21 NUMMELIN, E. 1984. General Irreducible Markov Chains and Non-Negative Operators. Cambridge Univ. Press.
  • 22 SHURENKOV, V. M. 1992. On the relationship between spectral radii and Perron roots. Technical report, Dept. Mathematics Goteborg Univ. ¨
  • 23 STROOCK, D. W. 1984. An Introduction to the Theory of Large Deviations. Springer, Berlin.
  • 24 VARADHAN, S. R. S. 1984. Large Deviations and Applications. SIAM, Philadelphia.
  • CLEVELAND, OHIO 44106 MADISON, WISCONSIN 53706 E-MAIL: ney@math.wisc.edu