I like to view Markov chain as simply state transition model. Let’s consider finite number of states to make thing simple.
- Time homogeneous: simply mean that the state transition matrix does not change over time
- Irreducible: I hate this term as I tend to forget what it really means. Model is irreducible if one can reach any state from any state. Can think of it as the states are connected “in probability”. I guess irreducible refer to the fact that the transition matrix cannot reduce to multiple submatrices to describe the same model
- Stationary (invariant) distribution (of the states): I tend to get confused of this with time homogenous. Or get confused with narrow/wide-sense stationary in stochastic process. Here it actually means that the probability does not change in the next iteration (i.e. $latex \pi =T \pi$, where $latex \pi$ is the stationary distribution and $latex T$ is the transition probability.
- Aperiodic: aperiodic if no perceivable period can exist. The greatest common divisor of all times revisiting the current state can only be one.