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Unformatted text preview: Chapter 4 One-Parameter Processes, Usually Functions of Time Section 4.1 defines one-parameter processes, and their variations (discrete or continuous parameter, one- or two- sided parameter), including many examples. Section 4.2 shows how to represent one-parameter 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 one-parameter processes, where the parameter is time in the first case and sample size in the second. 4.1 One-Parameter 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 (One-Parameter Process) A process whose index set T has one dimension is a one-parameter process . A process whose index set has more than one dimension is a multi-parameter process . A one-parameter process is 18 CHAPTER 4. ONE-PARAMETER PROCESSES 19 discrete or continuous depending on whether its index set is countable or un- countable. A one-parameter process where the index set has a minimal element, otherwise it is two-sided . N is a one-sided discrete index set, Z a two-sided discrete index set, R + (in- cluding zero!) is a one-sided continuous index set, and R a two-sided continuous index set. Most of this course will be concerned with one-parameter processes, which are intensely important in applications. This is because the one-dimensional 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 one-sided infinite sequence of independent, identically-distributed binary variables, where X t = 1 with probability p , for all t . Example 36 (Markov models) Markov chains are discrete-parameter stochas- tic processes. They may be either one-sided or two-sided. So are Markov models of order k , and hidden Markov models. Continuous-time Markov processes are, naturally enough, continuous-parameter stochastic processes, and again may be either one-sided or two-sided....
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- Spring '06