{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71257"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71257","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Some Limit Theorems (Empirical Processes)","abstract":"We establish a central limit theorem and bounded and compact laws of the iterated logarithm for partial sum processes indexed by classes of functions. For the central limit theorem, we assume an envelope condition and a majorizing measure condition more general than the usual metric entropy with bracketing. For the laws of the iterated logarithm, we assume an envelope condition and a growth condition on the metric entropy under bracketing. Examples show that our results are sharp. As a corollary we obtain new results for weighted sums of independent identically distributed random variables.","abstract_html":"We establish a central limit theorem and bounded and compact laws of the iterated logarithm for partial sum processes indexed by classes of functions. For the central limit theorem, we assume an envelope condition and a majorizing measure condition more general than the usual metric entropy with bracketing. For the laws of the iterated logarithm, we assume an envelope condition and a growth condition on the metric entropy under bracketing. Examples show that our results are sharp. As a corollary we obtain new results for weighted sums of independent identically distributed random variables.","abstract_has_math":false,"creators":["Lacey, Michael Thoreau"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:20Z","date_published":"2014-12-16T06:18:20Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8721684"],"render_values":[{"text":"(UMI)AAI8721684","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71257","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Lacey, Michael Thoreau"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:20Z","10000-01-01","1987"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71257","(UMI)AAI8721684"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We establish a central limit theorem and bounded and compact laws of the iterated logarithm for partial sum processes indexed by classes of functions. For the central limit theorem, we assume an envelope condition and a majorizing measure condition more general than the usual metric entropy with bracketing. For the laws of the iterated logarithm, we assume an envelope condition and a growth condition on the metric entropy under bracketing. Examples show that our results are sharp. As a corollary we obtain new results for weighted sums of independent identically distributed random variables.","Let X be a real-valued random variable with distribution function F(x) and characteristic function c(t). Let c$\\sb{\\rm n}$(t) be the characteristic function of F$\\sb{\\rm n}$(x), the nth empirical distribution function. We give necessary and sufficient conditions, in terms of c(t), for ${\\rm n\\sp{1/2}(c\\sb{n}(t)}$ $-$ c(t)) to obey bounded and compact laws of the iterated logarithm in C($-1,1$), the Banach space of continuous complex-valued functions on ($-1,1$).","Made available in DSpace on 2014-12-16T06:18:20Z (GMT). No. of bitstreams: 1 8721684.pdf: 2361088 bytes, checksum: 375ddb08dc6f607b79990935c42391d4 (MD5) Previous issue date: 1987","Embargo set by: Seth Robbins for item 71423 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","123 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987."]},{"key":"dc:title","label":"Title","values":["Some Limit Theorems (Empirical Processes)"]}]}],"canonical_facts":{"dc:creator":["Lacey, Michael Thoreau"],"dc:date":["2014-12-16T06:18:20Z","10000-01-01","1987"],"dc:description":["We establish a central limit theorem and bounded and compact laws of the iterated logarithm for partial sum processes indexed by classes of functions. For the central limit theorem, we assume an envelope condition and a majorizing measure condition more general than the usual metric entropy with bracketing. For the laws of the iterated logarithm, we assume an envelope condition and a growth condition on the metric entropy under bracketing. Examples show that our results are sharp. As a corollary we obtain new results for weighted sums of independent identically distributed random variables.","Let X be a real-valued random variable with distribution function F(x) and characteristic function c(t). Let c$\\sb{\\rm n}$(t) be the characteristic function of F$\\sb{\\rm n}$(x), the nth empirical distribution function. We give necessary and sufficient conditions, in terms of c(t), for ${\\rm n\\sp{1/2}(c\\sb{n}(t)}$ $-$ c(t)) to obey bounded and compact laws of the iterated logarithm in C($-1,1$), the Banach space of continuous complex-valued functions on ($-1,1$).","Made available in DSpace on 2014-12-16T06:18:20Z (GMT). No. of bitstreams: 1 8721684.pdf: 2361088 bytes, checksum: 375ddb08dc6f607b79990935c42391d4 (MD5) Previous issue date: 1987","Embargo set by: Seth Robbins for item 71423 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","123 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987."],"dc:identifier":["http://hdl.handle.net/2142/71257","(UMI)AAI8721684"],"dc:subject":["Mathematics"],"dc:title":["Some Limit Theorems (Empirical Processes)"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:04Z"}