h2 - is a simplex. Give an example where P % % FixPoint( M...

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Probability II MAP6473 4810 Homework-2. Prof. JLF King 4May2010 Reading. Please read examples 8.1 and 8.2 on P.112 of Billingsley, as well as 8.3 and 8.4. Note that Billingsley’s MChains may have countably - many states; hence there need not be an invariant distribution. Notation in force. Fix a positive integer D . Let P = P D - 1 be the simplex of probability vec- tors v R D . Fix a D × D (column)-stochastic matrix M ; each column is a probability vector. 2.1: Let K 0 := P . For each posint n let K n := M n ( P ) note ==== M ( K n - 1 ) . Then Λ := T 0 K n is compact and non-void. Prove that M () maps Λ into itself (not necessarily properly). Since each K n is convex, so is Λ. Prove or disprove:
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Unformatted text preview: is a simplex. Give an example where P % % FixPoint( M ) . By FixPoint( M ) I mean the set of M-invariant prob-ability vectors. 2.2: With as above: Prove or disprove: The M mapping sends onto itself. Optional. ( no points, other than brownie points ) For those who like Topology: Suppose f : X is a con-tinuous map on a compact metric space. Let := T n =0 f n ( X ). Give sucient and necessary conditions on ( X,f ) for f to map onto . Give examples of various possibilities....
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