Upcrossing inequalities for stationary sequences and applications
Michael Hochman
Source: Ann. Probab. Volume 37, Number 6
(2009), 2135-2149.
Abstract
For arrays (Si,j)1≤i≤j of random variables that are stationary in an appropriate sense, we show that the fluctuations of the process (S1,n)n=1∞ can be bounded in terms of a measure of the “mean subadditivity” of the process (Si,j)1≤i≤j. We derive universal upcrossing inequalities with exponential decay for Kingman’s subadditive ergodic theorem, the Shannon–MacMillan–Breiman theorem and for the convergence of the Kolmogorov complexity of a stationary sample.
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Keywords: Ergodic theorem; Shannon–McMillan–Breiman theorem; upcrossing inequalities entropy; Kolmogorov complexity; almost everywhere convergence
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1258380784
Digital Object Identifier: doi:10.1214/09-AOP460
Mathematical Reviews number (MathSciNet): MR2573553
Zentralblatt MATH identifier: 05708796
References
[1] Bishop, E. (1966). An upcrossing inequality with applications. Michigan Math. J. 13 1–13.
Mathematical Reviews (MathSciNet): MR194562
Digital Object Identifier: doi:10.1307/mmj/1028999474
Project Euclid: euclid.mmj/1028999474
[2] Bishop, E. (1967). Foundations of Constructive Analysis. McGraw-Hill, New York.
Mathematical Reviews (MathSciNet): MR221878
Zentralblatt MATH: 0183.01503
[3] Brudno, A. A. (1982). Entropy and the complexity of the trajectories of a dynamic system. Trudy Moskov. Mat. Obshch. 44 124–149.
Mathematical Reviews (MathSciNet): MR656285
[4] Collet, P. and Eckmann, J.-P. (1997). Oscillations of observables in 1-dimensional lattice systems. Math. Phys. Electron. J. 3 3 (electronic).
Mathematical Reviews (MathSciNet): MR1456389
Zentralblatt MATH: 0897.70004
[5] Cover, T. M., Gács, P. and Gray, R. M. (1989). Kolmogorov’s contributions to information theory and algorithmic complexity. Ann. Probab. 17 840–865.
[6] Doob, J. L. (1953). Stochastic Processes. Wiley, New York.
Mathematical Reviews (MathSciNet): MR58896
[7] Dubins, L. E. (1972). Some upcrossing inequalities for uniformly bounded martingales. In Symposia Mathematica, Vol. IX (Convegno di Calcolo delle Probabilità, INDAM, Rome, 1971) 169–177. Academic Press, London.
[8] Ivanov, V. V. (1996). Oscillations of averages in the ergodic theorem. Dokl. Akad. Nauk 347 736–738.
Mathematical Reviews (MathSciNet): MR1398765
[9] Jones, R. L., Kaufman, R., Rosenblatt, J. M. and Wierdl, M. (1998). Oscillation in ergodic theory. Ergodic Theory Dynam. Systems 18 889–935.
Mathematical Reviews (MathSciNet): MR1645330
Zentralblatt MATH: 0924.28009
Digital Object Identifier: doi:10.1017/S0143385798108349
[10] Kalikow, S. and Weiss, B. (1999). Fluctuations of ergodic averages. In Proceedings of the Conference on Probability, Ergodic Theory, and Analysis (Evanston, IL, 1997) 43 480–488.
Mathematical Reviews (MathSciNet): MR1700603
Zentralblatt MATH: 0935.28006
Project Euclid: euclid.ijm/1255985104
[11] Krawczak, M. (1985). On upcrossing inequalities for subadditive superstationary processes. Ergodic Theory Dynam. Systems 5 409–416.
Mathematical Reviews (MathSciNet): MR805838
Zentralblatt MATH: 0575.60005
Digital Object Identifier: doi:10.1017/S0143385700003035
[12] Shields, P. C. (1996). The Ergodic Theory of Discrete Sample Paths. Graduate Studies in Mathematics 13. Amer. Math. Soc., Providence, RI.
Mathematical Reviews (MathSciNet): MR1400225
Zentralblatt MATH: 0879.28031
[13] Walters, P. (1982). An Introduction to Ergodic Theory. Graduate Texts in Mathematics 79. Springer, New York.
Mathematical Reviews (MathSciNet): MR648108
[14] Weiss, B. (2003). Actions of amenable groups. In Topics in Dynamics and Ergodic Theory. London Mathematical Society Lecture Note Series 310 226–262. Cambridge Univ. Press, Cambridge.
Mathematical Reviews (MathSciNet): MR2052281
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