{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71197"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71197","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Inequalities for Random Walk and Partially Observed Brownian Motion","abstract":"This thesis is divided into two parts. The first part studies the control of the maximal function of N-dimensional Brownian motion, B(,t), by the maximal function of partially observed Brownian motion. Let R denote a fixed open subset of (//R)('N), G an arbitrary open subset, and T the first exit time of the Brownian motion from G. Define the maximal function, B(,T)('*), by","abstract_html":"This thesis is divided into two parts. The first part studies the control of the maximal function of N-dimensional Brownian motion, B(,t), by the maximal function of partially observed Brownian motion. Let R denote a fixed open subset of (//R)('N), G an arbitrary open subset, and T the first exit time of the Brownian motion from G. Define the maximal function, B(,T)('*), by","abstract_has_math":false,"creators":["Mcconnell, Terry 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":"2014-12-16T06:18:01Z","date_published":"2014-12-16T06:18:01Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8203525"],"render_values":[{"text":"(UMI)AAI8203525","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71197","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Mcconnell, Terry Robert"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:01Z","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":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71197","(UMI)AAI8203525"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This thesis is divided into two parts. The first part studies the control of the maximal function of N-dimensional Brownian motion, B(,t), by the maximal function of partially observed Brownian motion. Let R denote a fixed open subset of (//R)('N), G an arbitrary open subset, and T the first exit time of the Brownian motion from G. Define the maximal function, B(,T)('*), by","(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)","and the partially observed maximal function, B(,T)('*)(,(FDIAG)R), by","Let x(,0) be a fixed point not belonging to the closure of R and p a positive number. Then there is a constant C(,1) so that the inequality","(1) E('x(,0))(B(,T)('*))('p) (LESSTHEQ) C(,1)E('x(,0))(B(,T)('*)(,(FDIAG)R))('p)","holds as G varies provided that there exists a function u, harmonic in R, and constants C(,2) > 0 and q > p such that","(2) (VBAR)x(VBAR)('q) (LESSTHEQ) u(x) (LESSTHEQ) C(,2)(VBAR)x(VBAR)('q) + C(,2), x (ELEM) R.","Conversely, if (1) holds then so does (2) with q replaced by p. This result has applications in complex analysis and probability.","The second part considers the integrability of exit times of random walks in N-dimensions (N (GREATERTHEQ) 2). Let S(,n) = X(,1) + X(,2) + ... + X(,n) be the n('th) partial sum of independent, identically distributed random vectors, X(,1),X(,2),..., having mean zero, finite second moments, and covariance matrix equal to the identity. Let W be an open subset of (//R)('N) which is invariant under positive dilations, and (tau) the first exit time of S(,n) from W. If the boundary of W satisfies certain regularity conditions then the range of exponents p for which (tau)(' 1/2) has a finite p('th )moment is essentially the same as the corresponding range for standard Brownian motion.","Made available in DSpace on 2014-12-16T06:18:01Z (GMT). No. of bitstreams: 1 8203525.pdf: 1711883 bytes, checksum: 7c9ad495d7943e58432997772f6d1e74 (MD5) Previous issue date: 1981","Embargo set by: Seth Robbins for item 71363 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","64 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981."]},{"key":"dc:title","label":"Title","values":["Inequalities for Random Walk and Partially Observed Brownian Motion"]}]}],"canonical_facts":{"dc:creator":["Mcconnell, Terry Robert"],"dc:date":["2014-12-16T06:18:01Z","10000-01-01","1981"],"dc:description":["This thesis is divided into two parts. The first part studies the control of the maximal function of N-dimensional Brownian motion, B(,t), by the maximal function of partially observed Brownian motion. Let R denote a fixed open subset of (//R)('N), G an arbitrary open subset, and T the first exit time of the Brownian motion from G. Define the maximal function, B(,T)('*), by","(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)","and the partially observed maximal function, B(,T)('*)(,(FDIAG)R), by","Let x(,0) be a fixed point not belonging to the closure of R and p a positive number. Then there is a constant C(,1) so that the inequality","(1) E('x(,0))(B(,T)('*))('p) (LESSTHEQ) C(,1)E('x(,0))(B(,T)('*)(,(FDIAG)R))('p)","holds as G varies provided that there exists a function u, harmonic in R, and constants C(,2) > 0 and q > p such that","(2) (VBAR)x(VBAR)('q) (LESSTHEQ) u(x) (LESSTHEQ) C(,2)(VBAR)x(VBAR)('q) + C(,2), x (ELEM) R.","Conversely, if (1) holds then so does (2) with q replaced by p. This result has applications in complex analysis and probability.","The second part considers the integrability of exit times of random walks in N-dimensions (N (GREATERTHEQ) 2). Let S(,n) = X(,1) + X(,2) + ... + X(,n) be the n('th) partial sum of independent, identically distributed random vectors, X(,1),X(,2),..., having mean zero, finite second moments, and covariance matrix equal to the identity. Let W be an open subset of (//R)('N) which is invariant under positive dilations, and (tau) the first exit time of S(,n) from W. If the boundary of W satisfies certain regularity conditions then the range of exponents p for which (tau)(' 1/2) has a finite p('th )moment is essentially the same as the corresponding range for standard Brownian motion.","Made available in DSpace on 2014-12-16T06:18:01Z (GMT). No. of bitstreams: 1 8203525.pdf: 1711883 bytes, checksum: 7c9ad495d7943e58432997772f6d1e74 (MD5) Previous issue date: 1981","Embargo set by: Seth Robbins for item 71363 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","64 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981."],"dc:identifier":["http://hdl.handle.net/2142/71197","(UMI)AAI8203525"],"dc:subject":["Mathematics"],"dc:title":["Inequalities for Random Walk and Partially Observed Brownian Motion"],"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"}