# hw4 - X and p = p x y x,y ∈X a |X|×|X| matrix of...

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Stat 310A/Math 230A Theory of Probability Homework 4 Andrea Montanari Due on October 21, 2010 Solutions should be complete and concisely written. Please, use a separate sheet (or set of sheets) per each problem. Staple sheets referring to the same problem, and write your name on each sheet. You are welcome to discuss problems with your colleagues, but should write and submit your own solution. In some cases, multiple homework options will be proposed (and indicated as ‘Option 1’, ‘Option 2’, etc.). You are welcome to work on all the problems proposed (solutions will be posted), but should submit only those corresponding to one ‘Option’. Exercises on independent random variables and product measures Solve Exercises [1.4.18], [1.4.38], [1.4.39], [1.4.43] in Amir Dembo’s lecture notes. An exercise on Markov chains Let X be a ﬁnite set, q be a probability law over
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Unformatted text preview: X and p = { p ( x, y ) } x,y ∈X a |X|×|X| matrix of non-negative numbers such that, for any x ∈ X : X y ∈X p ( x, y ) = 1 . (1) A Markov chain is a probability measure over ( X N , F ), with X N = { ( ω , ω 1 , ω 2 , . . . ) : ω i ∈ X} and F the σ-algebra generated by cylindrical sets, such that P ( { ω : ω = x , ω 1 = x 1 . . . ω n = x n } ) = q ( x ) n-1 Y i =0 p ( x i , x i +1 ) , (2) for any n ≥ 0. Check that this indeed deﬁnes a probability distribution using Kolmogorov extension theorem. Let X i ( ω ) = ω i and recall that the tail σ-algebra is deﬁned by T ≡ ± n ≥ σ ( { X i } i ≥ n ) . (3) Prove that, if p ( x, y ) > 0 for any x, y ∈ X , then T is trivial. [Hint: It might be a good idea to remind yourself the statement of Perron-Frobenius theorem]...
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