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Unformatted text preview: 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 σfield and the notion of sample paths, and then redefines stochastic processes as random functions whose sample paths lie in nice sets. You will have seen, briefly, the definition of a stochastic process in 36752, 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 / Xmeasurable 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 subcollection of random variables. Example 2 Any single random variable is a (trivial) stochastic process. (Take T = { 1 } , say.) 2 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 onesided discrete (real, complex, vectorvalued, . . . ) random sequence. Most of the stochastic processes you have encountered are probably of this sort: Markov chains, discreteparameter martingales, etc. Example 5 Let T = Z and Ξ be as in Example 4. Then { X t } t ∈ T is a twosided random sequence. Example 6 Let T = Z d and Ξ be as in Example 4. Then { X t } t ∈ T is a d dimensional spatiallydiscrete random field. Example 7 Let T = R and Ξ = R . Then { X t } t ∈ T is a realvalued, continuous time random process (or random motion or random signal)....
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This note was uploaded on 12/20/2011 for the course STAT 36754 taught by Professor Schalizi during the Spring '06 term at University of Michigan.
 Spring '06
 Schalizi

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