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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