Unformatted text preview: re easily characterized, statistically, than general random processes, and they occur commonly in nature and engineering system. Kevin Buckley - 2007 Markov Sequences: Consider a discrete-time random sequence Xn . If, for all K and n, p(xn /xn−1 , xn−2 , · · · , xn−K ) = p(xn /xn−1 ) , then Xn is a Markov sequence. For a Markov sequence, 2 (3) p(xn , xn−1 , xn−2 , · · · , xn−K ) = p(xn /xn−1 ) p(xn−1 /xn−2 ) · · · p(xn−K +1 /xn−K ) p(xn−K )
K −1 = p(xn−K )
k =0 p(xn−k /xn−k−1) . (4) Vector Markov Sequences: To this point we have discussed only scalar Markov processes X (t) and sequences Xn . This discussion generalizes to vector random processes and sequences. For example, let X n denote an L-dimensional vector random sequence. X n is a vector Markov sequence if, for all K and n, p(xn /xn−1 , xn−2 , · · · , xn−K ) = p(xn /xn−1 ) . Then p(xn , xn−1 , xn−2 , · · · , xn−K ) = p(xn /xn−1 ) p(xn−1 /xn−2 ) · · · p(xn−K +1/xn−K ) p(xn−K )
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This document was uploaded on 10/12/2009.
- Spring '09