= Preservation under enlargement of transition support
Suppose two finite Markov chains on the same <state space> have transition <matrices> $P,Q$ satisfying
$$
P_{ij}>0\Longrightarrow Q_{ij}>0.
$$
Every positive-probability path and return cycle for $P$ is also possible for $Q$. Thus <irreducible Markov chain>[irreducibility] passes from $P$ to $Q$. For a state <aperiodic Markov chain>[aperiodic] under $P$, its positive return lengths under $Q$ contain those under $P$, so their <greatest common divisor> is still one. Hence <aperiodic Markov chain>[aperiodicity] also passes to $Q$.
Individual-state recurrence need not be preserved when the original chain is reducible. For example, take $P=I$ on $\{0,1\}$ and
$$
Q=\begin{pmatrix}1/2&1/2\\0&1\end{pmatrix}.
$$
State zero is <recurrent Markov chain>[recurrent] under $P$ but <transient Markov chain>[transient] under $Q$: once the chain leaves it, it never returns. The added transition opens an escape from its former closed class. If $P$ is finite and <irreducible Markov chain>[irreducible], however, then $Q$ is also finite and <irreducible Markov chain>[irreducible], so every state remains <positive recurrent Markov chain>[positive recurrent]. <mean recurrence time>[Mean recurrence times] can still change because the <Stationary distribution> depends on <transition probabilities>, not merely on their support.
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