License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2021.80
URN: urn:nbn:de:0030-drops-145204
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14520/
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Nandakumar, Satyadev ; Pulari, Subin

Ergodic Theorems and Converses for PSPACE Functions

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Abstract

We initiate the study of effective pointwise ergodic theorems in resource-bounded settings. Classically, the convergence of the ergodic averages for integrable functions can be arbitrarily slow [Ulrich Krengel, 1978]. In contrast, we show that for a class of PSPACE L¹ functions, and a class of PSPACE computable measure-preserving ergodic transformations, the ergodic average exists and is equal to the space average on every EXP random. We establish a partial converse that PSPACE non-randomness can be characterized as non-convergence of ergodic averages. Further, we prove that there is a class of resource-bounded randoms, viz. SUBEXP-space randoms, on which the corresponding ergodic theorem has an exact converse - a point x is SUBEXP-space random if and only if the corresponding effective ergodic theorem holds for x.

BibTeX - Entry

@InProceedings{nandakumar_et_al:LIPIcs.MFCS.2021.80,
  author =	{Nandakumar, Satyadev and Pulari, Subin},
  title =	{{Ergodic Theorems and Converses for PSPACE Functions}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{80:1--80:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14520},
  URN =		{urn:nbn:de:0030-drops-145204},
  doi =		{10.4230/LIPIcs.MFCS.2021.80},
  annote =	{Keywords: Ergodic Theorem, Resource-bounded randomness, Computable analysis, Complexity theory}
}

Keywords: Ergodic Theorem, Resource-bounded randomness, Computable analysis, Complexity theory
Collection: 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)
Issue Date: 2021
Date of publication: 18.08.2021

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