Order markov if pfxn1 in1 j xn in xn1 in1

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Unformatted text preview: solute values jX0 j; jX1 j; : : : to be a Markov chain? 1.6 Definition. We say that a process X0 ; X1 ; : : : is rth order Markov if PfXn+1 = in+1 j Xn = in ; Xn,1 = in,1 ; : : : ; X0 = i0 g = PfXn+1 = in+1 j Xn = in ; : : : ; Xn,r+1 = in,r+1 g for all n  r and all i0 ; : : : ; in+1 2 S. 1.7 Exercise A moving average process . Moving average models are used frequently in time series analysis, economics and engineering. For these models, one assumes that there is an underlying, unobserved process : : : ; Y,1 ; Y0 ; Y1 ; : : : of iid random variables. A moving average process takes an average possibly a weighted average of these iid random variables in a sliding window." For example, suppose that at time n we simply take the average of the Yn and Yn,1 , de ning Xn = 1=2Yn + Yn,1. Our goal is to show that the process X0 ; X1 ; : : : de ned in this way is not Markov. As a simple example, suppose that the distribution of the iid Y random variables is PfYi = 1g = 1=2 = PfYi = ,1g. 1. Show that X0 ; X1 ; : : : is not a Markov chain. 2. Show that X0 ; X1 ; : : : is not an rth order Markov chain for any nite r. We will use the shorthand Pi " to indicate a probability taken in a Markov chain started in state i at time 0. That is, Pi A" is shorthand for PfA j X0 = ig." We'll also use the notation E i " in an analogous way for expectation. 1.8 Notation. Stochastic Processes J. Chang, March 30, 1999 1. MARKOV CHAINS Page 1-6 Let fXn g be a nite-state Markov chain and let A be a subset of the state space. Suppose we want to determine the expected time until the chain enters the set A, starting from an arbitrary initial state. That is, letting A = inf fn  0 : Xn 2 Ag denote the rst time to hit A de ned to be 0 if X0 2 A , we want to determine E i  A . Show that 1.9 Exercise. E i  A = 1 + for i 2 A. = X k P i; kE k  A 1.10 Exercise. You are ipping a coin repeatedly. Which pattern would you expect to see faster: HH or HT? For example, if you get the sequence TTHHHTH..., then you see HH" at the 4th toss and HT" at the 6th. Letting N1 and N2 denote the times required to see HH" and HT", respectively, can you guess intuitively whether E N1  is smaller than, the same as, or larger than E N2 ? Go ahead, make a guess and my day . Why don't you also simulate some to see how the answer looks; I recommend a computer, but if you like tossing real coins, enjoy yourself by all means. Finally, you can use the reasoning of the Exercise 1.9 to solve the problem and evaluate E Ni . A hint is to set up a Markov chain having the 4 states HH, HT, TH, and TT. 1.11 Exercise. Here is a chance to practice formalizing some typical intuitively obvious" statements. Let X0 ; X1 ; : : : be a nite-state Markov chain. a. We start with an observation about conditional probabilities that will be a useful tool throughout the rest of this problem. Let F1 ; : : : ; Fm be disjoint events. Show that if S PE jFi  = p for all i = 1; : : : ; m then PE j...
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