{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71236"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71236","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"The Prediction Process of Step-Processes and Applications","abstract":"Frank B. Knight, in 1975, presented a new viewpoint in the theory of stochastic processes, by introducing what is called the prediction process. This dissertation applies that method to a special kind of","abstract_html":"Frank B. Knight, in 1975, presented a new viewpoint in the theory of stochastic processes, by introducing what is called the prediction process. This dissertation applies that method to a special kind of","abstract_has_math":false,"creators":["Goswami, Alok Kumar"],"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":"2014-12-16T06:18:14Z","date_published":"2014-12-16T06:18:14Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8600195"],"render_values":[{"text":"(UMI)AAI8600195","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71236","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Goswami, Alok Kumar"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:14Z","10000-01-01","1985"]},{"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/2142/71236","(UMI)AAI8600195"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Frank B. Knight, in 1975, presented a new viewpoint in the theory of stochastic processes, by introducing what is called the prediction process. This dissertation applies that method to a special kind of","process, namely step-processes. The prediction process is constructed for processes of the types (UNFORMATTED TABLE FOLLOWS)","(i) X(,t) = X (.) 1(,(t(GREATERTHEQ)T(,*))), 0 (LESSTHEQ) t (LESSTHEQ) (INFIN)(TABLE ENDS)","where X (NOT=) 0 and 0 (LESSTHEQ) T(,*) (LESSTHEQ) (INFIN) are random variables on a probability space, and, more generally,","(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)","where X(,0) (ELEM) (//R), J(,k) (NOT=) 0, 0 < S(,k) < (INFIN) are random variables on a probability space with","with probability 1 and, T(,0) = 0, T(,n) = S(,1) +...+ S(,n), for n (GREATERTHEQ) 1. The prediction process Z(,t) in either case becomes a Borel right process with unique left limits Z(,t-), and X(,t) is shown to be probabilistically equivalent to a Borel function of Z(,t). Discontinuities of Z(,t) are studied especially with regard to how they reflect the discontinuities of X(,t). A Levy system (N,H) for the Markov process Z(,t) is constructed and used to establish a known result on representation of martingales. Finally, the special case where the conditional distributions of waiting times for jumps given the past are exponential, is studied; and a charac- terisation of such processes in terms of the prediction process is derived. Also, a necessary and sufficient condition for the predic- tion process Z(,t) to be a step-process is proved; and, some simple examples (including non-markovian ones) are discussed.","Made available in DSpace on 2014-12-16T06:18:14Z (GMT). No. of bitstreams: 1 8600195.pdf: 4759103 bytes, checksum: f0d69c91758045df7c4909315eb3c149 (MD5) Previous issue date: 1985","Embargo set by: Seth Robbins for item 71402 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","168 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1985."]},{"key":"dc:title","label":"Title","values":["The Prediction Process of Step-Processes and Applications"]}]}],"canonical_facts":{"dc:creator":["Goswami, Alok Kumar"],"dc:date":["2014-12-16T06:18:14Z","10000-01-01","1985"],"dc:description":["Frank B. Knight, in 1975, presented a new viewpoint in the theory of stochastic processes, by introducing what is called the prediction process. This dissertation applies that method to a special kind of","process, namely step-processes. The prediction process is constructed for processes of the types (UNFORMATTED TABLE FOLLOWS)","(i) X(,t) = X (.) 1(,(t(GREATERTHEQ)T(,*))), 0 (LESSTHEQ) t (LESSTHEQ) (INFIN)(TABLE ENDS)","where X (NOT=) 0 and 0 (LESSTHEQ) T(,*) (LESSTHEQ) (INFIN) are random variables on a probability space, and, more generally,","(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)","where X(,0) (ELEM) (//R), J(,k) (NOT=) 0, 0 < S(,k) < (INFIN) are random variables on a probability space with","with probability 1 and, T(,0) = 0, T(,n) = S(,1) +...+ S(,n), for n (GREATERTHEQ) 1. The prediction process Z(,t) in either case becomes a Borel right process with unique left limits Z(,t-), and X(,t) is shown to be probabilistically equivalent to a Borel function of Z(,t). Discontinuities of Z(,t) are studied especially with regard to how they reflect the discontinuities of X(,t). A Levy system (N,H) for the Markov process Z(,t) is constructed and used to establish a known result on representation of martingales. Finally, the special case where the conditional distributions of waiting times for jumps given the past are exponential, is studied; and a charac- terisation of such processes in terms of the prediction process is derived. Also, a necessary and sufficient condition for the predic- tion process Z(,t) to be a step-process is proved; and, some simple examples (including non-markovian ones) are discussed.","Made available in DSpace on 2014-12-16T06:18:14Z (GMT). No. of bitstreams: 1 8600195.pdf: 4759103 bytes, checksum: f0d69c91758045df7c4909315eb3c149 (MD5) Previous issue date: 1985","Embargo set by: Seth Robbins for item 71402 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","168 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1985."],"dc:identifier":["http://hdl.handle.net/2142/71236","(UMI)AAI8600195"],"dc:subject":["Mathematics"],"dc:title":["The Prediction Process of Step-Processes and Applications"],"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"}