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# h1b - Probability II MAP6473 4810 Homework-1b State B goes...

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Probability II MAP6473 4810 Homework-1b Prof. JLF King 4May2010 Reading. Please get your copy of his text as soon as possible. Please read Sect.7, pages 92–108 by Monday, 19Jan. Please read part of Sect.8, pages 111–121 by Friday, 23Jan. 1b1: Fix a posint D . Let P = P D - 1 be the simplex of probability vectors v R D . Fix a D × D ( column )- stochastic matrix M ; each column is a prob.vec. Given a vector v P , define the Ces`aro average v N := A N ( v ) := 1 N X j [ 0 .. N ) M j v . Prove that lim N →∞ A N ( v ) exists, and is in P . [ Hint: See EE1 on Markov pamphlet. ] 1b2: A D × D Markov matrix M determines transition probabilities. Use τ ( A B ) to denote the transition prob from state A to B ( it is the A, B -entry in M ). A distribution σ () on the states ( i.e, an M -left-invariant col-vector ) determines a one-sided Markov process ~ Y = Y 0 Y 1 Y 2 . . . where the prob of Y 0 Y 1 . . . Y N equaling word w 0 w 1 . . . w N is the product σ ( w 0 ) τ ( w 0 w 1 ) τ ( w 1 w 2 ) · · · τ ( w N - 1 w N ) . : Invariance. Now suppose that σ () is an invariant
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