2.3 Homogeneous Markov Processes
Definition 2.3.1. A Markov process is time-homogeneous if its transition probabilities depend only on the length of the time interval, not on when it starts:
- in discrete time, \(P(X_{m + n} = j\mid X_m = i) = P(X_n = j\mid X_0 = i)\) for all \(m\);
- in continuous time, \(P(X(s + t) = j\mid X(s) = i) = P(X(t) = j\mid X(0) = i)\) for all \(s\geq 0\).
All the Markov processes in this course are homogeneous. A homogeneous discrete-time Markov process with a countable state space is a Markov chain, the subject of chapter 3; continuous-time processes such as the Poisson process and the birth and death processes follow in chapters 4 and 5.
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