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Unformatted text preview: Chapter 2 Building Infinite Processes from FiniteDimensional Distributions Section 2.1 introduces the finitedimensional distributions of a stochastic process, and shows how they determine its infinitedimensional distribution. Section 2.2 considers the consistency conditions satisfied by the finitedimensional distributions of a stochastic process, and the ex tension theorems (due to Daniell and Kolmogorov) which prove the existence of stochastic processes with specified, consistent finite dimensional distributions. 2.1 FiniteDimensional Distributions So, we now have X , our favorite Ξvalued stochastic process on T with paths in U . Like any other random variable, it has a probability law or distribution, which is defined over the entire set U . Generally, this is infinitedimensional. Since it is inconvenient to specify distributions over infinitedimensional spaces all in a block, we consider the finitedimensional distributions . Definition 22 (Finitedimensional distributions) The finitedimensional dis tributions of X are the the joint distributions of X t 1 , X t 2 , . . . X t n , t 1 , t 2 , . . . t n ∈ T , n ∈ N . You will sometimes see “FDDs” and “fidis” as abbreviations for “finitedimensional distributions”. Please do not use “fidis”. We can at least hope to specify the finitedimensional distributions. But we are going to want to ask a lot of questions about asymptotics, and global proper ties of sample paths, which go beyond any finite dimension, so you might worry 7 CHAPTER 2. BUILDING PROCESSES 8 that we’ll still need to deal directly with the infinitedimensional distribution. The next theorem says that this worry is unfounded; the finitedimensional dis tributions specify the infinitedimensional distribution (pretty much) uniquely. Theorem 23 Let X and Y be two Ξvalued processes on T with paths in U . Then X and Y have the same distribution iff all their finitedimensional distri butions agree. Proof : “Only if”: Since X and Y have the same distribution, applying the any given set of coordinate mappings will result in identicallydistributed random vectors, hence all the finitedimensional distributions will agree. “If”: We’ll use the π λ theorem. Let C be the finite cylinder sets, i.e., all sets of the form C = x ∈ Ξ T  ( x t 1 , x t 2 , . . . x t n ) ∈ B where n ∈ N , B ∈ X n , t 1 , t 2 , . . . t n ∈ T . Clearly, this is a πsystem, since it is closed under intersection. Now let L consist of all the sets L ∈ X T where P ( X ∈ L ) = P ( Y ∈ L ). We need to show that this is a λsystem, i.e., that it (i) includes Ξ T , (ii) is closed under complementation, and (iii) is closed under monotone increasing limits. (i) is clearly true: P ( X ∈ Ξ T ) = P ( Y ∈ Ξ T ) = 1. (ii) is true because we’re looking at a probability: if L ∈ L , then P ( X ∈ L c ) = 1 P ( X ∈ L ) = 1 P ( Y ∈ L ) = P ( Y ∈ L c ). To see (iii), let L n ↑ L be a monotoneincreasing...
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 Spring '06
 Schalizi
 Probability theory, Stochastic process, Probability space, ξt, FiniteDimensional Distributions

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