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Unformatted text preview: Chapter 2 Building Infinite Processes from Finite-Dimensional Distributions Section 2.1 introduces the finite-dimensional distributions of a stochastic process, and shows how they determine its infinite-dimensional distribution. Section 2.2 considers the consistency conditions satisfied by the finite-dimensional 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 Finite-Dimensional Distributions So, we now have X , our favorite Ξ-valued 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 infinite-dimensional. Since it is inconvenient to specify distributions over infinite-dimensional spaces all in a block, we consider the finite-dimensional distributions . Definition 22 (Finite-dimensional distributions) The finite-dimensional 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 “finite-dimensional distributions”. Please do not use “fidis”. We can at least hope to specify the finite-dimensional 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 infinite-dimensional distribution. The next theorem says that this worry is unfounded; the finite-dimensional dis- tributions specify the infinite-dimensional distribution (pretty much) uniquely. Theorem 23 Let X and Y be two Ξ-valued processes on T with paths in U . Then X and Y have the same distribution iff all their finite-dimensional 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 identically-distributed random vectors, hence all the finite-dimensional 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 π-system, 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 λ-system, 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 monotone-increasing...
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- Spring '06
- Probability theory, Stochastic process, Probability space, ξt, Finite-Dimensional Distributions