Unconditional Transition Probability

Topics: Markov Chain - Transition Probabilities in a Markov Chain

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(theorem)

The usual transition probabilities in a Markov chain are actually conditional in the sense that they consider a specific present state. For instance, in the case of one step transition probabilities, they are $\mathbb{P}\left(X_n=j\mid X_t=i\right)$.

If we want to obtain the unconditional transition probabilities, then we can use the initial probability vector $a=\left(a_0,a_1,\dots ,a_n\right)$ to calculate:

$$\mathbb{P}\left(X_n=j\right)=a_0{p}_{0j}^{(n)}+a_1{p}_{1j}^{(n)}+\cdots +a_m{p}_{mj}^{(n)}$$