Upcrossing inequalities for stationary sequences and applications
Abstract
For arrays $(S_{i,j})_{1\leq i\leq j}$ of random variables that are stationary in an appropriate sense, we show that the fluctuations of the process $(S_{1,n})_{n=1}^{\infty}$ can be bounded in terms of a measure of the ``mean subadditivity'' of the process $(S_{i,j})_{1\leq i\leq j}$. 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.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- August 2006
- DOI:
- 10.48550/arXiv.math/0608311
- arXiv:
- arXiv:math/0608311
- Bibcode:
- 2006math......8311H
- Keywords:
-
- Mathematics - Dynamical Systems;
- Mathematics - Probability;
- 37A30;
- 37A35;
- 60G10;
- 60G17;
- 94A17;
- 68Q30 (Primary)
- E-Print:
- Published in at http://dx.doi.org/10.1214/09-AOP460 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)