Deriving Transition Probabilities of a Markov Process

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I was reading this article about Markov Processes (https://www.jstatsoft.org/article/view/v066i06) and saw the following equation (4):

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I am trying to understand why equation (4) is true. I can understand where equation (2) comes from and where equation (3) comes from, but I am having difficulty in making the "leap" and understand the equivalency between equation (3) and equation (4).

  • Can someone please help me understand the relationship between (3) and (4)?

Thank you!