Upcrossing inequalities for stationary sequences and applications

Details Upcrossing inequalities for stationary sequences and applications Upcrossing inequalities for stationary sequences and applications

2006-08-13

arxiv.org/abs/math/0608311v3

Annals of Probability 2009, Vol. 37, No. 6, 2135-2149

Mathematics

Dynamical Systems

Published in at http://dx.doi.org/10.1214/09-AOP460 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/09-AOP460

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.

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