{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86833"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86833","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Qualitative and Quantitative Analysis of Weighted Ergodic Theorems","abstract":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Demeter, Ciprian"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Joseph M. 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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.","Made available in DSpace on 2015-09-28T15:19:46Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3153287.pdf: 4793015 bytes, checksum: 74a5ce0fb80ab91daeeb70461ad0ac23 (MD5) Previous issue date: 2004","Embargo set by: Seth Robbins for item 88114 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","118 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2004."]},{"key":"dc:title","label":"Title","values":["Qualitative and Quantitative Analysis of Weighted Ergodic Theorems"]}]}],"canonical_facts":{"dc:contributor":["Joseph M. Rosenblatt"],"dc:creator":["Demeter, Ciprian"],"dc:date":["2015-09-28T15:19:46Z","10000-01-01","2004"],"dc:description":["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.","Made available in DSpace on 2015-09-28T15:19:46Z (GMT). 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