×

zbMATH — the first resource for mathematics

On the convergence of averages of mixing sequences. (English) Zbl 0776.60035
Summary: We construct an absolutely regular stationary random sequence which is an instantaneous bounded function of an aperiodic recurrent Markov chain with a countable state space, such that the large deviation principle fails for the arithmetic means of the sequence, while the exponential convergence holds. We also show that exponential convergence holds for the arithmetic means of a vector valued strictly stationary bounded \(\varphi\)-mixing sequence.

MSC:
60F10 Large deviations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bradley, R. C. (1983). On the ?-mixing condition for stationary random sequences,Trans. Amer. Math. Soc. 276, 55-66. · Zbl 0514.60041
[2] Bradley, R. C. (1989). A stationary, pairwise independent absolutely regular sequence for which the central limit theorem fails,Probab. Th. Rel. Fields 81, 1-10. · Zbl 0649.60017 · doi:10.1007/BF00343735
[3] Bradley, R. C. (1986). Basic properties of strong mixing conditions. In:Dependence in Probability and Statistics (eds.), E. Eberlein and M. S. Taqqu, Birkh?user, Boston, pp. 165-192. · Zbl 0603.60034
[4] Bryc, W. (1992). On large deviations for uniformly strong mixing sequences,Stoch. Proc. Appl. 41, 191-202. · Zbl 0756.60027 · doi:10.1016/0304-4149(92)90120-F
[5] Bryc, W. (1992). On the large deviation principle for stationary weakly dependent random fields,Ann. Prob. 20, 1004-1030. · Zbl 0756.60028 · doi:10.1214/aop/1176989815
[6] Acosta, A. de (1990). Large deviations for empirical measures of Markov chains,J. Theoretical Prob. 3, 395-431. · Zbl 0711.60023 · doi:10.1007/BF01061260
[7] Doob, J. L. (1953).Stochastic Processes, Wiley, New York. · Zbl 0053.26802
[8] Harris, T. E. (1956). The existence of stationary measures for certain Markov processes,Third Berkeley Symposium on Mathematical Statistics and Probability, Vol. II, Berkeley, pp. 113-124. · Zbl 0072.35201
[9] Ibragimov, I. A., and Linnik, Y. V. (1971).Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff, Groningen. · Zbl 0219.60027
[10] Ney, P., and Nummelin, E. (1987). Markov additive process I. Eigenvalue properties and limit theorems.Ann. Prob. 15, 593-609. · Zbl 0625.60028 · doi:10.1214/aop/1176992160
[11] Orey, S., and Pelikan, S. (1988). Large deviation principles for stationary processes,Ann. Prob. 16, 1481-1495. · Zbl 0659.60051 · doi:10.1214/aop/1176991579
[12] Rosenblatt, M. (1971).Markov Processes. Structure and Asymptotic Behavion, Springer, Berlin. · Zbl 0236.60002
[13] Schonmann, R. H. (1989). Exponential Convergence Under Mixing,Probab. Theory Related Fields 81, 235-238. · doi:10.1007/BF00319552
[14] Varadhan, S. R. S. (1984).Large Deviations and Applications, SIAM, Philadelphia. · Zbl 0549.60023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.