Unformatted text preview: ‘ ∈ N (here we use the notation ω ‘ 1 = ( ω 1 ,...,ω ‘ )). Let B Ω be the Borel σalgebra associated to this topology. For n even, let A n be the set of sequences deﬁned as follows A n = n ω ∈ Ω : n X i =1 ω i = n/ 2 , ω i = 0 for all i > n o . (2) Consider the sequence of probability measures { ν n } n ∈ 2 N , with ν n the uniform distribution over A n , i.e. ν n ( { ω } ) = ( ³ n n/ 2 )1 if ω ∈ A n , otherwise. (3) 1. Show that, for each n , ν n is indeed a measure over B Ω . 2. What is the weak limit of ν n as n → ∞ ? Prove your answer. In solving the last point you can assume the following Fact 1. Let h : Ω → R be a continuous function. Then h is uniformly continuous in the following sense. There exists a function δ : N → R , ‘ 7→ δ ( ‘ ) , with lim ‘ →∞ δ ( ‘ ) = 0 , such that, for any ω ∈ Ω , ω ∈ N ‘ ( ω ) , we have  h ( ω )h ( ω )  ≤ δ ( ‘ ) ....
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 '11
 MONTANARI,A.
 Topology, Probability, Limit of a sequence, νn, Andrea Montanari, Stat 310A/Math 230A, multiple homework options

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