Upper-Lower Class Tests for Weighted i.i.d. Sequences and Martingales

István Berkes (), Siegfried Hörmann () and Michel Weber ()
Additional contact information
István Berkes: Graz University of Technology
Siegfried Hörmann: University of Utah
Michel Weber: Université Louis-Pasteur et C.N.R.S.

Journal of Theoretical Probability, 2010, vol. 23, issue 2, 428-446

Abstract: Abstract We study the upper-lower class behavior of weighted sums ∑ k=1 n a k X k , where X k are i.i.d. random variables with mean 0 and variance 1. In contrast to Feller’s classical results in the case of bounded X j , we show that the refined LIL behavior of such sums depends not on the growth properties of (a n ) but on its arithmetical distribution, permitting pathological behavior even for bounded (a n ). We prove analogous results for weighted sums of stationary martingale difference sequences. These are new even in the unweighted case and complement the sharp results of Einmahl and Mason obtained in the bounded case. Finally, we prove a general upper-lower class test for unbounded martingales, improving several earlier results in the literature.

Keywords: Weighted i.i.d. sequences; Upper-lower class tests; Law of the iterated logarithm; Martingales; 60F15; 60G50; 60G42 (search for similar items in EconPapers)
Date: 2010
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10959-009-0219-5 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:23:y:2010:i:2:d:10.1007_s10959-009-0219-5

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

DOI: 10.1007/s10959-009-0219-5

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 ().