{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

831_09hw2

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

This preview shows page 1. Sign up to view the full content.

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
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online