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Evidently, ν and µ have identical marginal distributions: ν$_{X}$$_{i}$ = µ$_{X}$$_{i}$ , and also identical joint distributions of any consecutive pairs: ν$_{X}$$_{i}$$_{,X}$$_{i}$$_{+1}$ = µ$_{X}$$_{i}$$_{,X}$$_{i}$$_{+1}$ . Therefore 

and 

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Since µ ∈ Λ( γ, q, c ) , we have ν ∈ Λ( γ, q, c ) . Let { X$_{i}$ } follow distribution µ and { Z$_{i}$ } follow distribution ν . Then 

<img src='content_image/298958.jpg'>

where equality holds if and only if { X$_{i}$ } is a first-order Markov process. By (11) and (105), L ( ν ) ≥ L ( µ ) . So for any µ which maximizes L ( µ ) , ν can be generated from µ by (99) with L ( ν ) ≥ L ( µ ) . L ( µ ) must be maximized by a first-order Markov process. 

Property (c): Suppose ν is a stationary fist-order Markov process, sufficiently denote as ν = {X , P$_{X}$$_{2}$$_{|}$$_{X}$$_{1}$ } , where X is the state space of ν and P$_{X}$$_{2}$$_{|}$$_{X}$$_{1}$ is the transition probability distribution. Define a new first-order Markov process ¯ ν from ν as follows. 

Definition 1: Let ¯ ν , defined on the same state space X as ν , be a first-order Markov process denoted by ( X , P$_{Z}$$_{2}$$_{|}$$_{Z}$$_{1}$ ) , where 

<img src='content_image/298954.jpg'>

