Theorem 5.3.2 — recurrence is contagious

If $x$ is recurrent and $\rho_{xy}>0$ — the chain started at $x$ has some chance of reaching $y$ — then $y$ is recurrent too, and moreover $\rho_{yx}=1$: from $y$ the chain returns to $x$ with probability one.

The second conclusion is the sharper one and it comes first in the proof. If $\rho_{yx}$ were less than one, then starting from $x$ the chain could reach $y$ and then fail ever to return, which would make $\rho_{xx}<1$ and contradict the recurrence of $x$. Once $\rho_{yx}=1$ is known, $y$ inherits recurrence: from $y$ the chain reaches $x$ surely, from $x$ it returns to $x$ surely and can reach $y$ again, and the return series at $y$ diverges with the one at $x$.

The consequence is that recurrence is a property of a communicating class, not of an individual state, which is what makes the decomposition of the state space into closed irreducible recurrent classes possible.

Formalization Note Both conclusions are asserted, since the second is what the first is proved from and is the more useful of the two. No assumption is made about $\rho_{yx}$ beyond what is derived.