{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71206"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71206","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Invariance Principles for Random Processes Generated by Extrema and Partial Sums of Random Variables","abstract":"Assume that F is a distribution function such that for some non-degenerate distribution function G, and sequences of constants a(,n) (a(,n) positive) and b(,n), we have that, as n increases, F('n)(a(,n)x + b(,n)) converges to G(x) for every real value x. Under these conditions we prove a strong invariance principle for extremal processes analogous to the well-known invariance principle of Strassen for partial sums. In our case, the strong invariance principle yields several weak convergence results for extremal processes. This differs from the case of partial sums in that Strassen's result does not yield the central limit theorem. By a similar procedure, we also obtain a strong invariance principle for point processes in the plane.","abstract_html":"Assume that F is a distribution function such that for some non-degenerate distribution function G, and sequences of constants a(,n) (a(,n) positive) and b(,n), we have that, as n increases, F('n)(a(,n)x + b(,n)) converges to G(x) for every real value x. Under these conditions we prove a strong invariance principle for extremal processes analogous to the well-known invariance principle of Strassen for partial sums. In our case, the strong invariance principle yields several weak convergence results for extremal processes. This differs from the case of partial sums in that Strassen's result does not yield the central limit theorem. By a similar procedure, we also obtain a strong invariance principle for point processes in the plane.","abstract_has_math":false,"creators":["Dabrowski, Andre Robert"],"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":"2014-12-16T06:18:04Z","date_published":"2014-12-16T06:18:04Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8302844"],"render_values":[{"text":"(UMI)AAI8302844","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71206","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Dabrowski, Andre Robert"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:04Z","10000-01-01","1982"]},{"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":["(UMI)AAI8302844","http://hdl.handle.net/2142/71206"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Assume that F is a distribution function such that for some non-degenerate distribution function G, and sequences of constants a(,n) (a(,n) positive) and b(,n), we have that, as n increases, F('n)(a(,n)x + b(,n)) converges to G(x) for every real value x. Under these conditions we prove a strong invariance principle for extremal processes analogous to the well-known invariance principle of Strassen for partial sums. In our case, the strong invariance principle yields several weak convergence results for extremal processes. This differs from the case of partial sums in that Strassen's result does not yield the central limit theorem. By a similar procedure, we also obtain a strong invariance principle for point processes in the plane.","Using a different method, we prove an invariance principle in probability for extremal processes arising from stationary sequences satisfying certain weak dependence conditions. Of independent interest is a generalization to dependent sequences of a technique of Major used to obtain an universal approximating sequence from a collection of sequences, each of which approximates the desired sequence only to within a fixed positive tolerance.","We improve a strong invariance principle for partial sums of a stationary (phi)-mixing sequence of random variables with finite (2 + (delta)) moment (0 < (delta) (LESSTHEQ) 2) due to Berkes and Philipp. The rate of decay, (phi)(n), can be as slow as log n to the power -(1 + (epsilon)) (1 + 2/(delta)), where (epsilon) is some positive value. The error term obtained is sufficient to yield the central limit theorem and upper and lower class refinements to the law of the iterated logarithm.","The generalization of a technique of Major mentioned earlier is used to improve an invariance principle in probability due to Philipp for partial sums of (phi)-mixing sequences of Banach space valued random variables in the domain of attraction to a Gaussian law.","Made available in DSpace on 2014-12-16T06:18:04Z (GMT). No. of bitstreams: 1 8302844.pdf: 2337324 bytes, checksum: cf92a8fe841f60ad1ed6d62fccb788ff (MD5) Previous issue date: 1982","Embargo set by: Seth Robbins for item 71372 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","101 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1982."]},{"key":"dc:title","label":"Title","values":["Invariance Principles for Random Processes Generated by Extrema and Partial Sums of Random Variables"]}]}],"canonical_facts":{"dc:creator":["Dabrowski, Andre Robert"],"dc:date":["2014-12-16T06:18:04Z","10000-01-01","1982"],"dc:description":["Assume that F is a distribution function such that for some non-degenerate distribution function G, and sequences of constants a(,n) (a(,n) positive) and b(,n), we have that, as n increases, F('n)(a(,n)x + b(,n)) converges to G(x) for every real value x. Under these conditions we prove a strong invariance principle for extremal processes analogous to the well-known invariance principle of Strassen for partial sums. In our case, the strong invariance principle yields several weak convergence results for extremal processes. This differs from the case of partial sums in that Strassen's result does not yield the central limit theorem. By a similar procedure, we also obtain a strong invariance principle for point processes in the plane.","Using a different method, we prove an invariance principle in probability for extremal processes arising from stationary sequences satisfying certain weak dependence conditions. Of independent interest is a generalization to dependent sequences of a technique of Major used to obtain an universal approximating sequence from a collection of sequences, each of which approximates the desired sequence only to within a fixed positive tolerance.","We improve a strong invariance principle for partial sums of a stationary (phi)-mixing sequence of random variables with finite (2 + (delta)) moment (0 < (delta) (LESSTHEQ) 2) due to Berkes and Philipp. The rate of decay, (phi)(n), can be as slow as log n to the power -(1 + (epsilon)) (1 + 2/(delta)), where (epsilon) is some positive value. The error term obtained is sufficient to yield the central limit theorem and upper and lower class refinements to the law of the iterated logarithm.","The generalization of a technique of Major mentioned earlier is used to improve an invariance principle in probability due to Philipp for partial sums of (phi)-mixing sequences of Banach space valued random variables in the domain of attraction to a Gaussian law.","Made available in DSpace on 2014-12-16T06:18:04Z (GMT). No. of bitstreams: 1 8302844.pdf: 2337324 bytes, checksum: cf92a8fe841f60ad1ed6d62fccb788ff (MD5) Previous issue date: 1982","Embargo set by: Seth Robbins for item 71372 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","101 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1982."],"dc:identifier":["(UMI)AAI8302844","http://hdl.handle.net/2142/71206"],"dc:subject":["Mathematics"],"dc:title":["Invariance Principles for Random Processes Generated by Extrema and Partial Sums of Random Variables"],"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"}