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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 -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 well 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 -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: Well use the - theorem. Let C be the finite cylinder sets, i.e., all sets of the form C = x 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 -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 -system, i.e., that it (i) includes T , (ii) is closed under complementation, and (iii) is closed under monotone increasing limits. (i) is clearly true: P ( X T ) = P ( Y T ) = 1. (ii) is true because were 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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This note was uploaded on 12/20/2011 for the course STAT 36-754 taught by Professor Schalizi during the Spring '06 term at University of Michigan.
- Spring '06