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87 a p1 t0 p fxt 2 a t e t tg the manipulation

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Unformatted text preview: oupling. Consider chains fXt g, fYt g having transition rule P , with X0   and Y0  . Look at PfX1 = Y1 g, conditioning on whether or not X0 = Y0 . *** ALSO apply this stu back to a Gibbs sampling example. 1.10.4 More about stationary distributions *** Omit or incorporate in earlier sections? Suppose the chain has a positive recurrent atom , so that E T  1. De ne A = E hPT ,1 i t=0 I fXt 2 Ag : T  What is this? Remember the I denotes an indicator random variable. The sum PT ,1 I fX 2 Ag is accumulating 0's and 1's as t ranges over the values 0; 1; : : : ; T , 1. t t=0 So the sum is simply a count of the number of times that Xt 2 A holds for t between 0 and T , 1. In other words, the sum is the number of visits made by X0 ; : : : ; XT ,1 to the set A, and the numerator of A is the expected number of such visits. Think again of the cycle" idea, where a cycle is now a portion of the Markov chain path between successive 1.86 Stochastic Processes E J. Chang, March 30, 1999 1.10. GENERAL STATE SPACE MARKOV CHAINS Page 1-47 visits to the state . Then A is the expected number of times the chain visits the set A during a cycle, divided by the expected length of a cycle. Now, Ta is a random variable, so the sum in 1.86 is running over a random number of terms. That looks a bit hard to work with, but we can use the following standard and useful trick, which should be your rst reaction when you see sums like this: we make the summation sign run over all possible t values and introduce another indicator function to restrict the sum to the values of t that we want. That is, TX1 , t=0 I fXt 2 Ag = 1 X t=0 I fXt 2 AgI ft Ta g = 1 X t=0 I fXt 2 A; Ta tg: Taking the expected value, since the expected value of an indicator random variable is its probability, we can write  in the equivalent form   A = 1.87 P1 t=0 P fXt 2 A; T E T  tg : The manipulation from 1.86 to 1.87 is so fundamental and often used in probability that you will often see it used without any comment. It is a trick that is well worth mastering and remembering. 1.88 Proposition. Let fXt g be a Markov chain with a positive recurrent atom , and de ne hPT ,1 i P1 P fX 2 A; T t=0 I fXt 2 Ag = t=0 E tT  A = E T  Then  is a stationary distribution for fXt g. E tg : R Proof: Clearly  is a probability distribution. We want to R show that P x; Adx = A. De ning A = E T A, we want to show that P x; Adx = A. We have Z But P x; Adx = 1 XZ t=0 P fXt 2 dx; T tgP x; A: P x; A = P fXt+1 2 A j Xt = xg = P fXt+1 2 A j Xt = x; T tg; where the last equality holds by the Markov property, because the event fT tg = c depends only on the random variables X0 ; : : : ; Xt . That is, given the precise fT  tg information about the state Xt = x, we can throw away the information T t. So Z P x; Adx = = Stochastic Processes 1 XZ t=0 1 X t=0 P P fXt 2 dx; T fXt+1 2 A; T tgP fXt+1 2 A j Xt = x; T tg tg J. Chang, March 30, 1999 Page 1-48 = E = "TX1 , E t=0 "X T t=1 R I...
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