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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 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
(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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