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Unformatted text preview: Chapter 4 OneParameter Processes, Usually Functions of Time Section 4.1 defines oneparameter processes, and their variations (discrete or continuous parameter, one or two sided parameter), including many examples. Section 4.2 shows how to represent oneparameter processes in terms of shift operators. Weve been doing a lot of pretty abstract stuff, but the point of this is to establish a common set of tools we can use across many different concrete situa tions, rather than having to build very similar, specialized tools for each distinct case. Today were going to go over some examples of the kind of situation our tools are supposed to let us handle, and begin to see how they let us do so. In particular, the two classic areas of application for stochastic processes are dy namics (systems changing over time) and inference (conclusions changing as we acquire more and more data). Both of these can be treated as oneparameter processes, where the parameter is time in the first case and sample size in the second. 4.1 OneParameter Processes The index set T isnt, usually, an amorphous abstract set, but generally some thing with some kind of topological or geometrical structure. The number of (topological) dimensions of this structure is the number of parameters of the process. Definition 34 (OneParameter Process) A process whose index set T has one dimension is a oneparameter process . A process whose index set has more than one dimension is a multiparameter process . A oneparameter process is 18 CHAPTER 4. ONEPARAMETER PROCESSES 19 discrete or continuous depending on whether its index set is countable or un countable. A oneparameter process where the index set has a minimal element, otherwise it is twosided . N is a onesided discrete index set, Z a twosided discrete index set, R + (in cluding zero!) is a onesided continuous index set, and R a twosided continuous index set. Most of this course will be concerned with oneparameter processes, which are intensely important in applications. This is because the onedimensional parameter is usually either time (when were doing dynamics) or sample size (when were doing inference), or both at once. There are also some important cases where the single parameter is space. Example 35 (Bernoulli process) You all know this one: a onesided infinite sequence of independent, identicallydistributed binary variables, where X t = 1 with probability p , for all t . Example 36 (Markov models) Markov chains are discreteparameter stochas tic processes. They may be either onesided or twosided. So are Markov models of order k , and hidden Markov models. Continuoustime Markov processes are, naturally enough, continuousparameter stochastic processes, and again may be either onesided or twosided....
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 Spring '06
 Schalizi

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