Stationary Transition Probability

Topics: Transition Probabilities in a Markov Chain - Markov Chain

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(definition)

Given a Markov chain $\left\{X_t\mid t\in \mathbb{N}\right\}$ and its one step transition probabilities, if, for every $t,i,j$, we have that:

$$\mathbb{P}(X_{t+1}=j\mid X_t=j)=\mathbb{P}(X_1=j\mid X_0=i)$$

…then we say the one step transition probabilities are stationary. That is, the transition probabilities don’t change across time and they are always the same.