Transcription of MARKOV CHAINS: BASIC THEORY
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MARKOV CHAINS: BASIC THEORY
MARKOV CHAINS: BASIC THEORY1. MA R KOVCH A INS A ND T H EIRTR A NSI T IO NPRO BA B ILIT I and First (discrete-time) MARKOV chain with (finite or countable) state spaceXis a se-quenceX0,X1, .. ofX valued random variables such that for all statesi,j,k0,k1, and alltimesn=0, 1, 2, .. ,(1)P(Xn+1=j Xn=i,Xn 1=kn 1, ..)=p(i,j)wherep(i,j)depends only on the statesi,j, and not on the timenor the previous stateskn 1,n 2, ..The numbersp(i,j)are called thetransition probabilitiesof the random walkon the integer latticeZdis the MARKOV chain whose tran-sition probabilities arep(x,x ei)=1/(2d) x Zdwheree1,e2.
A stochastic matrix is a square nonnegative matrix all of whose row sums are 1. A substochastic matrix is a square nonnegative matrix all of whose row sums are 1. A doubly stochastic matrix is a stochastic matrix all of whose column sums are 1. Observe that if you start with a stochastic matrix and delete the rows and columns indexed
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