Theorem 5.3.1 — recurrence is divergence of the return series

Let $N(y)$ be the number of visits the chain makes to $y$ at positive times, so that $\mathbb{E}_yN(y)=\sum_{n\ge1}p^n(y,y)$. Then

This is the bridge between the two ways of looking at a chain. Recurrence is defined by a hitting probability, $\rho_{yy}=1$, which is a statement about first passages; the criterion restates it as divergence of a series of matrix entries, which is computable. Almost every concrete recurrence verification — random walks in dimensions one, two and three among them — goes through this form.

The mechanism is the renewal identity $p^n(x,y)=\sum_{m\le n}f^m(x,y)\,p^{n-m}(y,y)$, which in generating-function form reads $P(s)=1/(1-F(s))$ at $x=y$: the return series diverges exactly when $F(1)=\rho_{yy}$ reaches one.

Formalization Note "$\sum_{n\ge1}p^n(y,y)=\infty$" is stated as failure of summability, which for a series of non-negative terms is exactly divergence, and avoids introducing extended reals. The visit count $N(y)$ is not itself formalized, since the chain's law on path space is not constructed; the statement is about the series, which is what $\mathbb{E}_yN(y)$ equals.