markov - Markovchains Jonathan L.F. King University of...

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Unformatted text preview: Markovchains Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA squash@ufl.edu Webpage http://www.math.ufl.edu/ squash/ 6 December, 2011 (at 11:08 ) Abstract: Markov chains, neither the 1-step nor the multi-step, are stable under finite-block codes. Geometric preliminaries. In a real vectorspace V , say that X N j =1 j v j ( with each j R ) : is a linear combination ( lin.comb ) of vectors ( points ) v 1 ,..., v N . If, further, these scalars satisfy 1 + 2 + + N = 1 , : then we call ( ) a weighted average of the points. Finally, if ( ) and each 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 affine-space 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 = 1 v 1 + 2 v 2 as a convex average ( of points v 1 , v 2 C ) , then necessarily v 1 = v 2 = w . A non-void set C V is an N-dimensional simplex ( an N-simplex ) if we can write it as C = Hull( w 1 ,..., w N +1 ) where no w j is in the affine-span of the others. Equivalently, C has precisely N +1 extreme-pts, 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 column-vector v . 1 : Perron-Frobenius Theorem (weak version) . There exists a fixpt P , i.e a column vector with M = . Proof (Brouwer fixed-pt) . Function M () is cts in, say, the L 1-topology. Since P is homeomorphic with the [ D- 1]-disk, Brouwer applies to yield a fixed-point 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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This note was uploaded on 01/26/2012 for the course MAS 4105 taught by Professor Rudyak during the Fall '09 term at University of Florida.

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markov - Markovchains Jonathan L.F. King University of...

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