Preservation under enlargement of transition support
= Preservation under enlargement of transition support
If two finite <Markov chains> satisfy
$$
P_{ij}>0\Longrightarrow Q_{ij}>0,
$$
then every positive-probability path and return cycle for $P$ also exists for $Q$. Consequently irreducibility and aperiodicity pass from $P$ to $Q$. Recurrence and mean return time need not do so, because the added transitions and changed probabilities can create escape routes or make returns less frequent.