{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71223"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71223","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"First Exit Times Through Curvilinear Boundaries for Stochastic Sequences","abstract":"Let [Special characters omitted.] be a stochastic sequence and let [Special characters omitted.] be a predictable sequence. Define S(,n) for each n by S(,n) = X(,0) + X(,1)V(,1) + ... + V(,n)X(,n). Let [Special characters omitted.] be an increasing sequence of positive numbers and let [Special characters omitted.] Let 0 (LESSTHEQ) (alpha) (LESSTHEQ) 2 and let L be a positive Borel function. For each n, define W(,n) by W(,n) = (VBAR)V(,n)(VBAR)('(alpha))L(b(,n)/(VBAR)V(,n)(VBAR)) if V(,n) (NOT=) 0, W(,n) = 0 if V(,n) = 0. Under certain conditions on the b(,n)'s involving (alpha) and under certain conditions on the first two conditional truncated moments of the V(,n)X(,n)'s involving (alpha) and W(,n)'s we show there exist positive constants q(,0), c(,0), c(,1), and c(,2) such that [Special characters omitted.] for all (lamda) > 0. Furthermore, q(,0), c(,0), c(,1), and c(,2) do not depend on [Special characters omitted.] except through the conditions mentioned above.","abstract_html":"Let [Special characters omitted.] be a stochastic sequence and let [Special characters omitted.] be a predictable sequence. Define S(,n) for each n by S(,n) = X(,0) + X(,1)V(,1) + ... + V(,n)X(,n). Let [Special characters omitted.] be an increasing sequence of positive numbers and let [Special characters omitted.] Let 0 (LESSTHEQ) (alpha) (LESSTHEQ) 2 and let L be a positive Borel function. For each n, define W(,n) by W(,n) = (VBAR)V(,n)(VBAR)('(alpha))L(b(,n)/(VBAR)V(,n)(VBAR)) if V(,n) (NOT=) 0, W(,n) = 0 if V(,n) = 0. Under certain conditions on the b(,n)'s involving (alpha) and under certain conditions on the first two conditional truncated moments of the V(,n)X(,n)'s involving (alpha) and W(,n)'s we show there exist positive constants q(,0), c(,0), c(,1), and c(,2) such that [Special characters omitted.] for all (lamda) > 0. Furthermore, q(,0), c(,0), c(,1), and c(,2) do not depend on [Special characters omitted.] except through the conditions mentioned above.","abstract_has_math":false,"creators":["Crabtree, James Claude"],"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":"2014-12-16T06:18:10Z","date_published":"2014-12-16T06:18:10Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8502115"],"render_values":[{"text":"(UMI)AAI8502115","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71223","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Crabtree, James Claude"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:10Z","10000-01-01","1984"]},{"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/2142/71223","(UMI)AAI8502115"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Let [Special characters omitted.] be a stochastic sequence and let [Special characters omitted.] be a predictable sequence. Define S(,n) for each n by S(,n) = X(,0) + X(,1)V(,1) + ... + V(,n)X(,n). Let [Special characters omitted.] be an increasing sequence of positive numbers and let [Special characters omitted.] Let 0 (LESSTHEQ) (alpha) (LESSTHEQ) 2 and let L be a positive Borel function. For each n, define W(,n) by W(,n) = (VBAR)V(,n)(VBAR)('(alpha))L(b(,n)/(VBAR)V(,n)(VBAR)) if V(,n) (NOT=) 0, W(,n) = 0 if V(,n) = 0. Under certain conditions on the b(,n)'s involving (alpha) and under certain conditions on the first two conditional truncated moments of the V(,n)X(,n)'s involving (alpha) and W(,n)'s we show there exist positive constants q(,0), c(,0), c(,1), and c(,2) such that [Special characters omitted.] for all (lamda) > 0. Furthermore, q(,0), c(,0), c(,1), and c(,2) do not depend on [Special characters omitted.] except through the conditions mentioned above.","When the (VBAR)V(,n)(VBAR)'s are nonrandom, the W(,n)'s are nonrandom so that our results give [Special characters omitted.] The conditions under which we prove these results are broad enough to include special cases for which [Special characters omitted.] is a sequence of i.i.d. random variables in the domain of attraction of a stable law and for which [Special characters omitted.] is a martingale difference sequence meeting the normed Marcinkiewicz-Zygmund conditions. The conditions on [Special characters omitted.] are rather general. For the lower bound on [Special characters omitted.] we require that [Special characters omitted.] For the upper bound, our conditions are met if W(,n)b(,n)('2-(alpha))/(b(,n)('2)-b(,n-1)('2)) is sufficiently large for each n.","We also use our results in the case b(,n) = b for each n (b some positive constant) to prove some a.s. convergence results.","Made available in DSpace on 2014-12-16T06:18:10Z (GMT). No. of bitstreams: 1 8502115.pdf: 2938101 bytes, checksum: 1bff9c6b3d3bd875484ba24d19159006 (MD5) Previous issue date: 1984","Embargo set by: Seth Robbins for item 71389 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","124 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1984."]},{"key":"dc:title","label":"Title","values":["First Exit Times Through Curvilinear Boundaries for Stochastic Sequences"]}]}],"canonical_facts":{"dc:creator":["Crabtree, James Claude"],"dc:date":["2014-12-16T06:18:10Z","10000-01-01","1984"],"dc:description":["Let [Special characters omitted.] be a stochastic sequence and let [Special characters omitted.] be a predictable sequence. Define S(,n) for each n by S(,n) = X(,0) + X(,1)V(,1) + ... + V(,n)X(,n). Let [Special characters omitted.] be an increasing sequence of positive numbers and let [Special characters omitted.] Let 0 (LESSTHEQ) (alpha) (LESSTHEQ) 2 and let L be a positive Borel function. For each n, define W(,n) by W(,n) = (VBAR)V(,n)(VBAR)('(alpha))L(b(,n)/(VBAR)V(,n)(VBAR)) if V(,n) (NOT=) 0, W(,n) = 0 if V(,n) = 0. Under certain conditions on the b(,n)'s involving (alpha) and under certain conditions on the first two conditional truncated moments of the V(,n)X(,n)'s involving (alpha) and W(,n)'s we show there exist positive constants q(,0), c(,0), c(,1), and c(,2) such that [Special characters omitted.] for all (lamda) > 0. Furthermore, q(,0), c(,0), c(,1), and c(,2) do not depend on [Special characters omitted.] except through the conditions mentioned above.","When the (VBAR)V(,n)(VBAR)'s are nonrandom, the W(,n)'s are nonrandom so that our results give [Special characters omitted.] The conditions under which we prove these results are broad enough to include special cases for which [Special characters omitted.] is a sequence of i.i.d. random variables in the domain of attraction of a stable law and for which [Special characters omitted.] is a martingale difference sequence meeting the normed Marcinkiewicz-Zygmund conditions. The conditions on [Special characters omitted.] are rather general. For the lower bound on [Special characters omitted.] we require that [Special characters omitted.] For the upper bound, our conditions are met if W(,n)b(,n)('2-(alpha))/(b(,n)('2)-b(,n-1)('2)) is sufficiently large for each n.","We also use our results in the case b(,n) = b for each n (b some positive constant) to prove some a.s. convergence results.","Made available in DSpace on 2014-12-16T06:18:10Z (GMT). No. of bitstreams: 1 8502115.pdf: 2938101 bytes, checksum: 1bff9c6b3d3bd875484ba24d19159006 (MD5) Previous issue date: 1984","Embargo set by: Seth Robbins for item 71389 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","124 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1984."],"dc:identifier":["http://hdl.handle.net/2142/71223","(UMI)AAI8502115"],"dc:subject":["Mathematics"],"dc:title":["First Exit Times Through Curvilinear Boundaries for Stochastic Sequences"],"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"}