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134 CHAPTER 4. THE LUCAS MODELAppendix—Markov ChainsLet X t be a random variable and x t be a particular realization of X t . AMarkov chain is a stochastic process {X t } ∞ t=0 with the property that theinformation in the current realized value of X t = x t summarizes the entirepast history of the process. That is,P[X t+1 = x t+1 |X t = x t ,X t−1 = x t−1 ,...,X 0 = x 0 ]=P[X t+1 = x t+1 |X t = x t ].(4.64)AkeyresultthatsimpliÞes probability calculations of Markov chains is,(98)⇒ Property 1 If {X t } ∞ t=0P[X t = x t ∩ X t−1 = x t−1 ∩ ···∩ X 0 = x 0 ]=is a Markov chain, thenP[X t = x t |X t−1 = x t−1 ] ···P[X 1 = x 1 |X 0 = x 0 ]P[X 0 = x 0 ]. (4.65)Proof: Let A j be the event (X j = x j ). You can write the left side of(4.65) as,t−1 \ t−1 \P(A t ∩ A t−1 ∩ ···∩ A 0 ) = P(A t | A j )P( A j ) (multiplication rule)j=0j=0j=0t−1 \= P(A t |A t−1 )P( A j ) (Markov chain property)t−2 \ t−2 \= P(A t |A t−1 )P(A t−1 | A j )P( A j ) (mult. rule)j=0j=0t−2 \= P(A t |A t−1 )P(A t−1 |A t−2 )P( ) (Markov chain)j=0.= P(A t |A t−1 )P(A t−1 |A t−2 ) ···P(A 1 |A 0 )P(A 0 )Let λ j , j = 1,...,N denote the possible states for X t .AMarkovchainhas stationary probabilities if the transition probabilities from state λ i to λ jare time-invariant. That is,P[X t+1 = λ j |X t = λ i ]=p ij