I have this Matrix:
$$P=\begin{pmatrix} 0 & 1 \\ 0.3 & 0.7 \end{pmatrix}$$
this markov chain is said to be aperiodic, I dont understand how it comes to it. Period $\delta$ is the gcd of the set of all diagonal elements, right? if $\delta>1$, $P$ is periodic, if $\delta=1$, then aperiodic.
but here it is not $\delta=1$, is it? or do i have to transit the matrix to some certain form?
Best Answer
Since my comment provided sufficient clarification:
When there's a stationary state, your system will evolve towards that state. In your case, the two left eigenvectors are $(−1,1)$ and $(3,10)$ with corresponding eigenvalues $−0.3$ and $1$. Every other state of the system can be decomposed into those two states. The first state exihibits oscillating behaviour, but it dies out as $0.3<1$. The other state is stationary. So whatever your initial state, you'll evolve towards that stationary state.