2.3 Homogeneous Markov Processes

1.

\(T\) is continuous.
Let \(\, 0 < t_0 < t_n < t\). If \(P\left (X(t) = j\, |\, X(t_n) = i\right )\, = \, P\left (X(t - t_n) = j\, |\, X(t_0) = i\right ).\) Then the process \(\{X(t)\}\) is a homogeneous Markov process.

2.

\(T\) is discrete. If \( P\left (X_{n + m} = j\, |\, X_m = i\right ) \, = \, P\left (X_n = j\, |\, X_0 = i\right )\). Then the Markovian process \(\, \{X_n\, , \, n = 0\, , \, 1\, , \, 2\, , \, \cdots \cdots \}\,\) is homogeneous.

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