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Journal of Probability and Statistics
Volume 2017, Article ID 9139645, 5 pages
Research Article

An Approach of Randomness of a Sample Based on Its Weak Ergodic Limit

Departamento de Matemáticas y Estadística, Universidad Nacional de Colombia, Manizales, Colombia

Correspondence should be addressed to Jaime A. Londoño; moc.liamg@onodnol.a.emiaj

Received 7 May 2017; Revised 19 September 2017; Accepted 24 October 2017; Published 5 November 2017

Academic Editor: Ramón M. Rodríguez-Dagnino

Copyright © 2017 Jaime A. Londoño. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


For a Polish Sample Space with a Borel -field with a surjective measurable transformation, we define an equivalence relation on sample points according to their ergodic limiting averages. We show that this equivalence relation partitions the subset of sample points on measurable invariant subsets, where each limiting distribution is the unique ergodic probability measure defined on each set. The results obtained suggest some natural objects for the model of a probabilistic time-invariant phenomenon are uniquely ergodic probability spaces. As a consequence of the results gained in this paper, we propose a notion of randomness that is weaker than recent approaches to Schnorr randomness.