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A Doubly stochastic matrix is a stochastic matrix M of which the columns also sum to 1, i.e., Mij 20, Mij=1, Mij=1, Vi, je S. jes a) Suppose that (Xn: n No) is a Markov Chain with state space S= {1,2,..., N) and with transition matrix P doubly stochastic. Show that the row vector with entries for all jES is a stationary distribution for {X,,:ne No}. IES b) Show that P - Q is doubly stochastic if both PERNXN and Q ERNXN are doubly stochastic matrices. c) Consider a MC with S= {1,2,3} and P= i) Is this chain time reversible? ii) Does this chain have a pointwise limiting distribution? If yes, provide this distribution.

A Doubly stochastic matrix is a stochastic matrix M of which the columns also sum to 1, i.e., Mij 20, Mij=1, Mij=1, Vi, je S. jes a) Suppose that (Xn: n No) is a Markov Chain with state space S= {1,2,..., N) and with transition matrix P doubly stochastic. Show that the row vector with entries for all jES is a stationary distribution for {X,,:ne No}. IES b) Show that P - Q is doubly stochastic if both PERNXN and Q ERNXN are doubly stochastic matrices. c) Consider a MC with S= {1,2,3} and P= i) Is this chain time reversible? ii) Does this chain have a pointwise limiting distribution? If yes, provide this distribution.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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A Doubly stochastic matrix is a stochastic matrix M of which the columns also sum to 1, i.e., Mij 20, Mij = 1, Mij = 1, Vi, je S. jeS ieS a) Suppose that {Xn :n € No} is a Markov Chain with state space S = {1,2,..., N} and with transition matrix P doubly stochastic. Show that the row vector v with entries v, + for all je S is a stationary distribution for {X, : ne No}. b) Show that P.Q is doubly stochastic if both P e RNXN and Q E RNXN are doubly stochastic matrices. c) Consider a MC with S {1,2,3} and TO P= TO i) Is this chain time reversible? ii) Does this chain have a pointwise limiting distribution? If yes, provide this distribution. d) Now consider a MC with S = {0, 1,2, 3} and 10 TO 5. TO 10 10 i) Find all stationary distributions of this chain. ii) What can you say about the long run behavior of this chain?
Transcribed Image Text:A Doubly stochastic matrix is a stochastic matrix M of which the columns also sum to 1, i.e., Mij 20, Mij = 1, Mij = 1, Vi, je S. jeS ieS a) Suppose that {Xn :n € No} is a Markov Chain with state space S = {1,2,..., N} and with transition matrix P doubly stochastic. Show that the row vector v with entries v, + for all je S is a stationary distribution for {X, : ne No}. b) Show that P.Q is doubly stochastic if both P e RNXN and Q E RNXN are doubly stochastic matrices. c) Consider a MC with S {1,2,3} and TO P= TO i) Is this chain time reversible? ii) Does this chain have a pointwise limiting distribution? If yes, provide this distribution. d) Now consider a MC with S = {0, 1,2, 3} and 10 TO 5. TO 10 10 i) Find all stationary distributions of this chain. ii) What can you say about the long run behavior of this chain?
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