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Unformatted text preview: Markovchains Jonathan L.F. King University of Florida, Gainesville FL 326112082, USA [email protected] Webpage http://www.math.ufl.edu/ ∼ squash/ 6 December, 2011 (at 11:08 ) Abstract: Markov chains, neither the 1step nor the multistep, are stable under finiteblock codes. Geometric preliminaries. In a real vectorspace V , say that X N j =1 α j v j ( with each α j ∈ R ) † : is a linear combination ( lin.comb ) of vectors ( points ) v 1 ,..., v N . If, further, these scalars satisfy α 1 + α 2 + ··· + α N = 1 , ‡ : then we call ( † ) a weighted average of the points. Finally, if ( ‡ ) and each α j > 0, then we call ( † ) a convex average of the points. Given a set S ⊂ V of points, we define three supersets Spn( S ) ⊃ AffSpn( S ) ⊃ Hull( S ) . The span is the set of all lin.combs ( † ), as { v 1 ,..., v N } ranges over all finite subsets of S . The affine span is the set of all ( † ) sat isfying ( ‡ ), whereas the hull is the smaller set of all convex aver ages. Thus Spn( S ) is the smallest subspace ( that includes S ) whereas AffSpn( S ) is the smallest affinespace and Hull( S ) is the smallest convex set . A point w ∈ C is an “ extreme point of a convex set C ” if: Whenever we write w = α 1 v 1 + α 2 v 2 as a convex average ( of points v 1 , v 2 ∈ C ) , then necessarily v 1 = v 2 = w . A nonvoid set C ⊂ V is an Ndimensional simplex ( an “ Nsimplex ” ) if we can write it as C = Hull( w 1 ,..., w N +1 ) where no w j is in the affinespan of the others. Equivalently, C has precisely N +1 extremepts, and Dim( C ) = N . Existence of an invariant vector Fix a posint D . Let P = P D 1 be the simplex of probability vectors v ∈ R D . Fix a D × D ( col umn )stochastic matrix M ; each column is a prob.vec. Let M : P denote the map v 7→ M v for a columnvector v . 1 : PerronFrobenius Theorem (weak version) . There exists a fixpt σ ∈ P , i.e a column vector σ with M σ = σ . ♦ Proof (Brouwer fixedpt) . Function M () is cts in, say, the L 1topology. Since P is homeomorphic with the [ D 1]disk, Brouwer applies to yield a fixedpoint σ ∈ P . Proof (Ces` aro averages) . Fix a vector v ∈ P . Let v N := A N ( v ) := 1 N X j ∈ [ ..N ) M j v ....
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 Fall '09
 RUDYAK
 Trigraph, Convex set, Prof. JLF King

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