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Title:  The Prediction Process of StepProcesses and Applications 
Author(s):  Goswami, Alok Kumar 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
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 of process, namely stepprocesses. 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 stepprocess is proved; and, some simple examples (including nonmarkovian ones) are discussed. 
Issue Date:  1985 
Type:  Text 
Description:  168 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1985. 
URI:  http://hdl.handle.net/2142/71236 
Other Identifier(s):  (UMI)AAI8600195 
Date Available in IDEALS:  20141216 
Date Deposited:  1985 
This item appears in the following Collection(s)

Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois