831_09hw2

831_09hw2 - that whenever we specify consistent probability...

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831 Theory of Probability Fall 2009 Homework 2 Due Tuesday, September 22 1. Let { X n,i : n N ,i ∈ I} be real-valued random variables on (Ω , F ,P ). Assume that for each n N , the variables { X n,i : i ∈ I} are independent. Assume that for each i ∈ I there is a real-valued random variable Y i on , F ,P ) such that X n,i Y i a.s. as n → ∞ . Show that { Y i : i ∈ I} are independent. (Hint: you may find it convenient to consider random variables of the type f ( X n,i ) where f is a bounded, continuous function.) 2. Let ( X n ) n 1 be a sequence of i.i.d. random variables with values in some measurable space ( S, A ). Let A ∈ A be a set such that P ( X 1 A ) > 0. Let T = inf { n 1 : X n A } be the index of the first sample that lies in the set A . Find the distribu- tions of T , X T and X T +1 . What can you say about the independence of ( T,X T ,X T +1 )? 3. Inspired by the Kolmogorov Extension Theorem, we might think
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Unformatted text preview: that whenever we specify consistent probability measures μ n on increasing σ-algebras A n , then we get a probability measure on the generated σ-algebra A = σ ( S n A n ). But consider this example. Let the space be the open unit interval (0 , 1). For each n ∈ N , let A n be the smallest σ-algebra that contains the set (0 , 1 n ) and the Borel subsets of [ 1 n , 1). On the measurable space ((0 , 1) , A n ) define the probability measure μ n by μ n { (0 , 1 n ) } = 1, μ n { [ 1 n , 1) } = 0. Show that the σ-algebras A n generate the Borel σ-algebra B (0 , 1) , but there is no probability measure μ on ((0 , 1) , B (0 , 1) ) such that, for all n , the restriction of μ to A n is μ n ....
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This note was uploaded on 11/18/2009 for the course MATH 241 taught by Professor Reed during the Spring '09 term at Duke.

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