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lecture-01 - Chapter 1 Basic Denitions Indexed Collections...

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Chapter 1 Basic Definitions: Indexed Collections and Random Functions Section 1.1 introduces stochastic processes as indexed collections of random variables. Section 1.2 builds the necessary machinery to consider random functions, especially the product σ -field and the notion of sample paths, and then re-defines stochastic processes as random functions whose sample paths lie in nice sets. You will have seen, briefly, the definition of a stochastic process in 36-752, but it’ll be useful to review it here. We will flip back and forth between two ways of thinking about stochastic processes: as indexed collections of random variables, and as random functions. As always, assume we have a nice base probability space ( Ω , F , P ), which is rich enough that all the random variables we need exist. 1.1 So, What Is a Stochastic Process? Definition 1 (Stochastic Process: As Collection of Random Variables) A stochastic process { X t } t T is a collection of random variables X t , taking val- ues in a common measure space ( Ξ , X ) , indexed by a set T . That is, for each t T , X t ( ω ) is an F / X -measurable function from Ω to Ξ , which induces a probability measure on Ξ in the usual way. It’s sometimes more convenient to write X ( t ) in place of X t . Also, when S T , X s or X ( S ) refers to that sub-collection of random variables. Example 2 Any single random variable is a (trivial) stochastic process. (Take T = { 1 } , say.) 2
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CHAPTER 1. BASICS 3 Example 3 Let T = { 1 , 2 , . . . k } and Ξ = R . Then { X t } t T is a random vector in R k . Example 4 Let T = { 1 , 2 , . . . } and Ξ be some finite set (or R or C or R k . . . ). Then { X t } t T is a one-sided discrete (real, complex, vector-valued, . . . ) random sequence. Most of the stochastic processes you have encountered are probably of this sort: Markov chains, discrete-parameter martingales, etc. Example 5 Let T = Z and Ξ be as in Example 4. Then { X t } t T is a two -sided random sequence. Example 6 Let T = Z d and Ξ be as in Example 4. Then { X t } t T is a d - dimensional spatially-discrete random field. Example 7 Let T = R and Ξ = R . Then { X t } t T is a real-valued, continuous- time random process (or random motion or random signal). Vector-valued processes are an obvious generalization. Example 8 Let T = B , the Borel field on the reals, and Ξ = R + , the non- negative extended reals. Then { X t } t T is a random set function on the reals.
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