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# ch1summary - 1.10 CHAPTER SUMMARY 1.10 55 Chapter Summary A...

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1.10. CHAPTER SUMMARY 55 1.10 Chapter Summary A Markov chain with transition probability p is defined by the property that given the present state the rest of the past is irrelevant for predicting the future: P ( X n +1 = y | X n = x, X n - 1 = x n - 1 , . . . , X 0 = x 0 ) = p ( x, y ) The m step transition probability p m ( i, j ) = P ( X n + m = y | X n = x ) is the m th power of the matrix p . Recurrence and transience The first thing we need to determine about a Markov chain is which states are recurrent and which are transient. To do this we let T y = min { n 1 : X n = y } and let ρ xy = P x ( T y < ) When x = y this is the probability X n ever visits y starting at x . When x = y this is the probability X n returns to y when it starts at y . We restrict to times n 1 in the definition of T Y so that we can say: y is recurrent if ρ yy = 1 and transient if ρ yy < 1. Transient states in a finite state space can all be identified using Theorem 1.3. If ρ xy > 0 , but ρ yx = 0 , then x is transient. Once the transient states are removed we can use Theorem 1.4. If C is a finite closed and irreducible set, then all states in C are recurrent. Here A is closed if x A and y A implies p ( x, y ) = 0, and B is irreducible if x, y B implies ρ xy > 0. The keys to the proof of Theorem 1.4 are that: (i) If x is recurrent and ρ xy > 0 then y is recurrent, and (ii) In a finite closed set there has to be at least one recurrent state. To prove these results, it was useful to know that if

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