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MARKOV CHAINS: BASIC THEORY - University of Chicago

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.

Then by the Chapman-Kolmogorov equa-tions and the law of total probability, P fXn =j g= X i (i)pn(i,j), equivalently, if the initial distribution is T (here we are viewing probability distributions on X as row vectors) then the distribution after n steps is T Pn. Notice that if there is a probability distribution on Xsuch that T = T P, then T ...

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