## Entropy and the uniform mean ergodic theorem for a family of sets

HTML articles powered by AMS MathViewer

- by Terrence M. Adams and Andrew B. Nobel PDF
- Trans. Amer. Math. Soc.
**369**(2017), 605-622 Request permission

## Abstract:

We define the entropy of an infinite family $\mathcal {C}$ of measurable sets in a probability space, and show that a family has zero entropy if and only if it is totally bounded under the symmetric difference semi-metric. Our principal result is that the mean ergodic theorem holds uniformly for $\mathcal {C}$ under every ergodic transformation if and only if $\mathcal {C}$ has zero entropy. When the entropy of $\mathcal {C}$ is positive, we establish a strong converse showing that the uniform mean ergodic theorem fails generically in every isomorphism class, including the isomorphism classes of Bernoulli transformations. As a corollary of these results, we establish that every strong mixing transformation is uniformly strong mixing on $\mathcal {C}$ if and only if the entropy of $\mathcal {C}$ is zero, and we obtain a corresponding result for weak mixing transformations.## References

- Terrence M. Adams and Andrew B. Nobel,
*Uniform approximation of Vapnik-Chervonenkis classes*, Bernoulli**18**(2012), no. 4, 1310–1319. MR**2995797**, DOI 10.3150/11-BEJ379 - Terrence M. Adams and Andrew B. Nobel,
*Uniform convergence of Vapnik-Chervonenkis classes under ergodic sampling*, Ann. Probab.**38**(2010), no. 4, 1345–1367. MR**2663629**, DOI 10.1214/09-AOP511 - Terrence M. Adams and Andrew B. Nobel,
*The gap dimension and uniform laws of large numbers for ergodic processes*, Preprint, 2010, arXiv:1007.2964v1. - Terrence Adams and Cesar E. Silva,
*$\textbf {Z}^d$ staircase actions*, Ergodic Theory Dynam. Systems**19**(1999), no. 4, 837–850. MR**1709423**, DOI 10.1017/S0143385799133923 - Patrice Assouad,
*Densité et dimension*, Ann. Inst. Fourier (Grenoble)**33**(1983), no. 3, 233–282 (French, with English summary). MR**723955** - J. R. Blum and D. L. Hanson,
*On the mean ergodic theorem for subsequences*, Bull. Amer. Math. Soc.**66**(1960), 308–311. MR**118803**, DOI 10.1090/S0002-9904-1960-10481-8 - R. M. Dudley,
*Uniform central limit theorems*, Cambridge Studies in Advanced Mathematics, vol. 63, Cambridge University Press, Cambridge, 1999. MR**1720712**, DOI 10.1017/CBO9780511665622 - Adam Fieldsteel and N. A. Friedman,
*Restricted orbit changes of ergodic $\textbf {Z}^d$-actions to achieve mixing and completely positive entropy*, Ergodic Theory Dynam. Systems**6**(1986), no. 4, 505–528. MR**873429**, DOI 10.1017/S0143385700003667 - Paul R. Halmos,
*Measure Theory*, D. Van Nostrand Co., Inc., New York, N. Y., 1950. MR**0033869** - Paul R. Halmos,
*Lectures on ergodic theory*, Publications of the Mathematical Society of Japan, vol. 3, Mathematical Society of Japan, Tokyo, 1956. MR**0097489** - Ulrich Krengel,
*On the speed of convergence in the ergodic theorem*, Monatsh. Math.**86**(1978/79), no. 1, 3–6. MR**510630**, DOI 10.1007/BF01300052 - V. A. Rokhlin,
*A general measure preserving transformation is not mixing*, Dokl. Akad. Nauk**60**(1948), 349-351. - J. Rosenblatt,
*Optimal norm approximation in ergodic theory*, Preprint, 2014. - Daniel J. Rudolph,
*The second centralizer of a Bernoulli shift is just its powers*, Israel J. Math.**29**(1978), no. 2-3, 167–178. MR**485401**, DOI 10.1007/BF02762006 - S. V. Tikhonov,
*A complete metric on the set of mixing transformations*, Mat. Sb.**198**(2007), no. 4, 135–158 (Russian, with Russian summary); English transl., Sb. Math.**198**(2007), no. 3-4, 575–596. MR**2352364**, DOI 10.1070/SM2007v198n04ABEH003850 - Aad W. van der Vaart and Jon A. Wellner,
*Weak convergence and empirical processes*, Springer Series in Statistics, Springer-Verlag, New York, 1996. With applications to statistics. MR**1385671**, DOI 10.1007/978-1-4757-2545-2 - Ramon van Handel,
*The universal Glivenko-Cantelli property*, Probab. Theory Related Fields**155**(2013), no. 3-4, 911–934. MR**3034796**, DOI 10.1007/s00440-012-0416-5 - Vladimir N. Vapnik,
*The nature of statistical learning theory*, 2nd ed., Statistics for Engineering and Information Science, Springer-Verlag, New York, 2000. MR**1719582**, DOI 10.1007/978-1-4757-3264-1

## Additional Information

**Terrence M. Adams**- Affiliation: U. S. Government, 9800 Savage Road, Ft. Meade, Maryland 20755
- MR Author ID: 338702
- Email: tmadam2@tycho.ncsc.mil
**Andrew B. Nobel**- Affiliation: Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-3260
- MR Author ID: 326596
- Email: nobel@email.unc.edu
- Received by editor(s): March 31, 2014
- Received by editor(s) in revised form: January 13, 2015
- Published electronically: March 21, 2016
- Additional Notes: The second author was supported by NSF Grants DMS-0907177 and DMS-1310002
- Journal: Trans. Amer. Math. Soc.
**369**(2017), 605-622 - MSC (2010): Primary 37A25; Secondary 60F05, 37A35, 37A50
- DOI: https://doi.org/10.1090/tran/6675
- MathSciNet review: 3557787