 Preservation under enlargement of transition support

Suppose two finite Markov chains on the same state space have transition matrices $P,Q$ satisfyingEvery positive-probability path and return cycle for $P$ is also possible for $Q$. Thus irreducibility passes from $P$ to $Q$. For a state aperiodic under $P$, its positive return lengths under $Q$ contain those under $P$, so their greatest common divisor is still one. Hence aperiodicity also passes to $Q$.

Individual-state recurrence need not be preserved when the original chain is reducible. For example, take $P=I$ on $\{0,1\}$ andState zero is recurrent under $P$ but 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, however, then $Q$ is also finite and irreducible, so every state remains positive recurrent. Mean recurrence times can still change because the Stationary distribution depends on transition probabilities, not merely on their support.