Finiteness of sum of Markov random variables till recurrence time

Question

Consider a positive recurrent Markov chain $X_n, n\geq 0$ on the set of strictly positive integers. Let the initial state of the Markov process be 1, i.e., $X_0 = 1$. Let $T$ denote the first recurrence time to state 1,i.e., $T = \inf\{n\geq 1:X_n = 1\}$. Is $\mathbb{E}[\sum_{k=0}^{T}X_k]< \infty$.

Answer 1

I believe that there is a counterexample. Let $\nu$ be a probability measure on $\Bbb N$ with infinite first moment (i.e. $\sum k\nu(k)=+\infty$), such that $\nu(k)>0$ for all $k \in \Bbb N$.

Define out transition kernels by $p(1,j) = \nu(j)$ and $p(i,j) = 1_{[j=1]}$ for $i \geq 2$.

This chain is positive recurrent because clearly $T \leq 2$ almost surely, and because all states communicate with each other.

However, $X_1$ has law $\nu$ and thus $E[X_1]=\infty$. And since $\sum_1^T X_j \geq X_1$ almost surely, the claim follows immediately.