Stochastic processes j chang march 30 1999 110 general

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Unformatted text preview: r some t  0g 0 for all x 2 S 2 For all states x 2 R and all subsets A S, P x; A  A. Conditions 1 and 2 pull in opposite directions: Roughly speaking, 1 wants the set R to be large, while 2 wants R to be small. Condition 1 requires that R be accessible from each state x 2 S. For example, 1 is satis ed trivially by taking R to be the whole state space S, but in that case 2 becomes a very demanding condition, asking for P x;    to hold for all states x 2 S. On the other hand, 2 is satis ed trivially if we take R to be any singleton fx1 g: just take  to be P x1 ;  and take = 0:9, for example. But in many examples each singleton is hit with probability 0, so that no singleton choice for R will satisfy condition 1. A Harris chain is one for which there is a set R that is simultaneously large enough to satisfy 1 but small enough to satisfy 2. Let's think a bit about the interpretation of 2. What does this inequality tell us? Writing P x; A =  P x; A , A  =: A + 1 , Qx; A; A + 1 ,  1, we have expressed the distribution P x;  as a mixture of two probability distributions and Qx; , where Qx;  is de ned by Qx; A = P x; A , A =1 , . Note that Qx;  is indeed a probability measure; for example, Qx; A  0 by the assumption that P x; A  A, and Qx; S = 1 because we have divided by the appropriate quantity 1 ,  in the de ning Qx; . Thus, we can simulate a draw from the distribution P x;  by the following procedure. Flip a coin" having Pheads = and Ptails = 1 , . If the outcome is heads, take a random draw from the distribution . If the outcome is tails, take a draw from the distribution Qx; . Stochastic Processes J. Chang, March 30, 1999 1.10. GENERAL STATE SPACE MARKOV CHAINS Page 1-45 It is useful to imagine essentially the same process in another slightly di erent way, on a slightly di erent state space. Let us adjoin an additional state, , to the given state space ~ S, obtaining the new state space S = S f g. This new state will be our accessible atom. We will say that the new chain visits the state whenever the old chain enters the set R and the coin ip turns up heads. Thus, after the state is entered, we know that the next state will be distributed according to the distribution ; note that this distribution is the same for all x 2 R. When the chain enters the state x 2 R and the coin ip turns up tails, the next state is chosen according to the distribution Qx; . ~ ~ To put all of this together, consider a Markov chain X0 ; X0 ; X1 ; X1 ; : : : generated recursively as follows. Suppose we are at time t, and we have already generated the value of ~ ~ Xt , and we are about to generate Xt . If Xt 2 Rc = S , R, then Xt = Xt . If Xt 2 R, then ~ we toss a coin. If the toss comes up heads, which happens with probability , then Xt = . ~ t = Xt . Next we use the value of Xt to generate Xt+1 . If ~ If the toss comes up tails, then X ~ ~ Xt = then Xt+1 is chosen from the distribution . If Xt 2 R then Xt...
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