Suppose I have a connected graph $G = (V,E)$, and a subset of vertices $U$, such that $W$ is the subgraph of $G$ induced by the vertices $V \setminus U$, and that $W$ is also connected.
Now, let $X_n$ be a random walk on $G$ and let $P$ be its transition probability matrix. Now, let $Q$ be the submatrix of $P$ that corresponds to the vertices $W$. So, $Q$ is a sub-stochastic matrix (i.e. at least $1$ row sums to less than $1$)
I have shown that for any $k,j \in W$, $(K^n)_{kj} = P_k(\{X_n = j\} \cap \{X_i \in W, \forall i < n\})$, where $P_k$ refers to the probability with respect to the random walk started at $k$.
I want to show that, for all $k, j \in W$, $$ \sum_{n=0}^\infty (K^n)_{kj} < \infty,$$ and also that $$ \sum_{n=0}^\infty x^n(K^n)_{kj}$$ has a radius of convergence strictly greater than $1$.
I can show both using a lot of linear algebra, most of which comes down to showing that $K$ has spectral radius less than $1$.
I was wondering if there is any probability arguments I can use to show this instead, as the solutions I have now seem extremely non-illustrative. Moreover, we haven't learned Perron-Frobenius and other similar results, which to me indicates that there must be a slicker, probabilistic solution to this problem.