The discrete Chapman-Kolmogorov equation

In deriving this equation, the following question is considered W/tat is the probability of transition of a system from state Sj to state Sk in exactly n steps i.e.  

In other words, pjk(n), the n-step transition probability function, is the conditional probability of occupying Sk at the nth step, given that the system initially occupied Sj. pjk(n), termed also higher transition probability, extends the one-step transition probability pjk(l) = Pjk and gives an answer to question 2 in 2.1-2. Note also that the function given by Eq.(2-26) is independent of t, since we are concerned in homogeneous transition probabilities. 

In answering the above question, we refer again to The Lost Jockey depicted in Fig.2-2 defined as system. We designate now point O, the initial state of the 

Noting that prob Sj Sj Sk) = prob Sj Sj and Sj - Sk, we may now have expressions for computing the probabilities of the transitions to the states listed in Eq.(2-27). Since the transitions to and from the Z states in Eq.(2-27) are mutually exclusive (that is, no pair of them can be occupied simultaneously), the probability of the transition from state j to state k in two steps, i.e. pjk(2), is equal to the sum of the probabilities over the Z different paths it is given by  

It should be noted that the above result follows also from the concept of conditional probability given by Eq.(2-3). 

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