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 Title: The Prediction Process of Step-Processes and Applications Author(s): Goswami, Alok Kumar Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Mathematics 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 ofprocess, 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(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)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. Issue Date: 1985 Type: Text Description: 168 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1985. URI: http://hdl.handle.net/2142/71236 Other Identifier(s): (UMI)AAI8600195 Date Available in IDEALS: 2014-12-16 Date Deposited: 1985
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