{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/68186"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/68186","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Some Almost Sure Convergence Results","abstract":"The first chapter of the thesis consists of an almost sure invariance principle for random variables in the Domain of Attraction of a Stable Law. Let {Y,Y(,1),Y(,2),...} be a sequence of i.i.d. symmetric random variables with Y in the domain of attraction of X where X is symmetric and stable of index (alpha). Suppose {a(,n)}, 0 ) 0 as i (--->) (INFIN). This extends an analogous result of Stout's where the more restrictive assumption is made that Y is in the domain of normal attraction of X.","abstract_html":"The first chapter of the thesis consists of an almost sure invariance principle for random variables in the Domain of Attraction of a Stable Law. Let {Y,Y(,1),Y(,2),...} be a sequence of i.i.d. symmetric random variables with Y in the domain of attraction of X where X is symmetric and stable of index (alpha). Suppose {a(,n)}, 0 ) 0 as i (--->) (INFIN). This extends an analogous result of Stout's where the more restrictive assumption is made that Y is in the domain of normal attraction of X.","abstract_has_math":false,"creators":["Fisher, Evan David"],"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":"2014-12-14T13:09:51Z","date_published":"2014-12-14T13:09:51Z","updated_at":"2026-07-22T22:25:58Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8203458"],"render_values":[{"text":"(UMI)AAI8203458","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/68186","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Fisher, Evan David"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-14T13:09:51Z","10000-01-01","1981"]},{"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":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/68186","(UMI)AAI8203458"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The first chapter of the thesis consists of an almost sure invariance principle for random variables in the Domain of Attraction of a Stable Law. Let {Y,Y(,1),Y(,2),...} be a sequence of i.i.d. symmetric random variables with Y in the domain of attraction of X where X is symmetric and stable of index (alpha). Suppose {a(,n)}, 0 ) 0 as i (--->) (INFIN). This extends an analogous result of Stout's where the more restrictive assumption is made that Y is in the domain of normal attraction of X.","The second chapter of the thesis contains an upper class law of the iterated logarithm for supermartingales, with hypotheses analogous to the Kolmogorov Law of the Iterated Logarithm. Let {U(,n), (,n), n (GREATERTHEQ) 1} be a supermartingale with X(,n) = U(,n) - U(,n-1). Define","(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)","and (theta)(,n) = (2 log(,2)s(,n)('2))(' 1/2) a.s. for n (GREATERTHEQ) 1 with s(,n) (--->) (INFIN) a.s. Suppose X(,i) (LESSTHEQ) K(,i)s(,i)/(theta)(,i) a.s. for each i (GREATERTHEQ) 1 where","K(,i) is (,i-1)-measurable and K (GREATERTHEQ) 1/2. A function (epsilon)((.)) is given so that","This result extends one of Stout's where he assumes 0 ) 0 as K (--->) 0 thus containing the Kolmogorov Law of the Iterated Logarithm as a special case.","In the third chapter of the thesis, two theorems are proved concerning normed weighted averages of a sequence of i.i.d. random variables. Let {Y,Y(,i), i (GREATERTHEQ) 1} be a sequence of i.i.d. random variables and let a(,j) > 0 with","Made available in DSpace on 2014-12-14T13:09:51Z (GMT). No. of bitstreams: 1 8203458.pdf: 1354353 bytes, checksum: 576ec98d98ac2bb1aca9dd6997b086f9 (MD5) Previous issue date: 1981","Embargo set by: Seth Robbins for item 68364 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","66 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981."]},{"key":"dc:title","label":"Title","values":["Some Almost Sure Convergence Results"]}]}],"canonical_facts":{"dc:creator":["Fisher, Evan David"],"dc:date":["2014-12-14T13:09:51Z","10000-01-01","1981"],"dc:description":["The first chapter of the thesis consists of an almost sure invariance principle for random variables in the Domain of Attraction of a Stable Law. Let {Y,Y(,1),Y(,2),...} be a sequence of i.i.d. symmetric random variables with Y in the domain of attraction of X where X is symmetric and stable of index (alpha). Suppose {a(,n)}, 0 ) 0 as i (--->) (INFIN). This extends an analogous result of Stout's where the more restrictive assumption is made that Y is in the domain of normal attraction of X.","The second chapter of the thesis contains an upper class law of the iterated logarithm for supermartingales, with hypotheses analogous to the Kolmogorov Law of the Iterated Logarithm. Let {U(,n), (,n), n (GREATERTHEQ) 1} be a supermartingale with X(,n) = U(,n) - U(,n-1). Define","(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)","and (theta)(,n) = (2 log(,2)s(,n)('2))(' 1/2) a.s. for n (GREATERTHEQ) 1 with s(,n) (--->) (INFIN) a.s. Suppose X(,i) (LESSTHEQ) K(,i)s(,i)/(theta)(,i) a.s. for each i (GREATERTHEQ) 1 where","K(,i) is (,i-1)-measurable and K (GREATERTHEQ) 1/2. A function (epsilon)((.)) is given so that","This result extends one of Stout's where he assumes 0 ) 0 as K (--->) 0 thus containing the Kolmogorov Law of the Iterated Logarithm as a special case.","In the third chapter of the thesis, two theorems are proved concerning normed weighted averages of a sequence of i.i.d. random variables. Let {Y,Y(,i), i (GREATERTHEQ) 1} be a sequence of i.i.d. random variables and let a(,j) > 0 with","Made available in DSpace on 2014-12-14T13:09:51Z (GMT). No. of bitstreams: 1 8203458.pdf: 1354353 bytes, checksum: 576ec98d98ac2bb1aca9dd6997b086f9 (MD5) Previous issue date: 1981","Embargo set by: Seth Robbins for item 68364 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","66 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981."],"dc:identifier":["http://hdl.handle.net/2142/68186","(UMI)AAI8203458"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["Some Almost Sure Convergence Results"],"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":"2026-07-22T22:25:58Z"}