Prokhorov Blocks and Strong Law of Large Numbers Under Rearrangements

S. Chobanyan (), S. Levental () and V. Mandrekar ()
Additional contact information
S. Chobanyan: Georgian Academy of Sciences
S. Levental: Michigan State University
V. Mandrekar: Michigan State University

Journal of Theoretical Probability, 2004, vol. 17, issue 3, 647-672

Abstract: Abstract We find conditions on a sequence of random variables to satisfy the strong law of large numbers (SLLN) under a rearrangement. It turns out that these conditions are necessary and sufficient for the permutational SLLN (PSLLN). By PSLLN we mean that the SLLN holds under almost all simple permutations within blocks the lengths of which grow exponentially (Prokhorov blocks). In the case of orthogonal random variables it is shown that Kolmogorov's condition, that is known not to be sufficient for SLLN, is actually sufficient for PSLLN. It is also shown that PSLLN holds for sequences that are strictly stationary with finite first moments. In the case of weakly stationary sequences a Gaposhkin result implies that SLLN and PSLLN are equivalent. Finally we consider the case of general norming and generalization of the Nikishin theorem. The methods of proof uses on the one hand the idea of Prokhorov blocks and Garsia's construction of product measure on the space of simple permutations, and on the other hand, a maximal inequality for permutations.

Keywords: Strong law of large numbers; permutations; Prokhorov blocks (search for similar items in EconPapers)
Date: 2004
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1023/B:JOTP.0000040292.52298.fd Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:17:y:2004:i:3:d:10.1023_b:jotp.0000040292.52298.fd

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/10959

DOI: 10.1023/B:JOTP.0000040292.52298.fd

Access Statistics for this article

Journal of Theoretical Probability is currently edited by Andrea Monica

More articles in Journal of Theoretical Probability from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().