I have a markov chain with the following $Q$ matrix:
$q_{n,n+1}= 1$, $q_{n, 0} = \delta_n$ for $n\neq0$. That is, for any $n\geq1$ we move to $n+1$ with rate $\delta_n$ or to $0$ with rate $\delta_n$, and we move to $1$ to $0$ with rate 1 with probability 1.
I want to show that this chain is transient iff $\sum_n \delta_n<\infty$. I tried looking at the jump chain: $p_{n,n+1} = \frac{1}{1+\delta_n}$, and $p_{n,0}=\frac{\delta_n}{1+\delta_n}$. Naturally I thought that the result $\sum_np_{0,0}(n)<\infty \iff \text{chain is transient}$ would be useful (since it is irreducible), but I don't think it is - I'm finding it difficult to get an expression for $p_{0,0}(n)$.
Any ideas?
