Qualitative and Quantitative Analysis of Weighted Ergodic Theorems

The second part of the thesis is devoted to the qualitative analysis of weighted operators, where the weights are obtained by almost everywhere sampling in a stationary stochastic process. The major theme of our investigation is whether one can break the duality, in other words, if one still gets convergence if the duality restriction is removed. Given the difficulty of these questions, we try to get a better understanding of their analogue with deterministic weights. A particularly interesting issue that was open for some time, concerns the validity of the weighted ergodic theorem with Besicovitch weights. This, again, was known to hold in the duality range; however, we prove here that the result fails for each pair of nondual indices. The positive results we obtain are about stationary sequences with finite first moment. In this spirit, we get some positive new results on the almost everywhere convergence of weighted one-sided series in L1. Also, we prove a best possible result for weighted series of integrable i.i.d.'s with random weights.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Demeter, Ciprian

Contributors dc:contributor

• Joseph M. Rosenblatt

• Mathematics

Language dc:language

eng

Identifier

(MiAaPQ)AAI3153287

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/86833