Claim: Birth and death chains have a stationary distribution if and only if $$\sum_m\prod_{j=1}^{m}\frac{p_{j-1}}{q_j}<\infty$$
Why is this true? I know that if the state space is $S=\{0,1,..\}$ with $p(i,i+1)=p_i$, $p(i,i-1)=1-p_i=q_i$ and $p_i,q_i>0$, then the chain is irreducible and hence all the states are recurrent if there is a stationary distribution. Also, the chain is recurrent if and only if $$\sum_m\prod_{j=1}^{m}\frac{q_j}{p_j}=\infty$$
So may I know why the latter equation implies the former? Intuitively, the former is close to the sum of reciprocal of the product in the latter equation which is a divergent series but am not sure. Thanks.
If a birth and death chain has a stationary distribution, the chain must satisfy the detailed balance equations
$$\pi\left(i\right)p\left(i,\ i-1\right)=\pi\left(i-1\right)p\left(i-1,\ i\right)$$
$$\pi\left(i\right)q_{i}=\pi\left(i-1\right)p_{i-1}$$
From this, we can formulate that $$\pi\left(i\right)=\pi\left(i-1\right)\cdot\frac{p_{i-1}}{q_{i}}$$
Now we know that $\pi\left(i-1\right)=\pi\left(i-2\right)\cdot\frac{p_{i-2}}{q_{i-1}}, \pi\left(i-2\right)=\pi\left(i-3\right)\cdot\frac{p_{i-3}}{q_{i-2}}... $
Thus, $\pi\left(m\right)=\pi\left(0\right)\prod_{j=0}^{m}\frac{p_{j-1}}{q_{j}}$, where $\pi\left(0\right)$ is some constant. In order for the stationary distribution $\pi\left(m\right)$ to be a valid probability distribution, it must satisfy $\sum_{m=0}^{\infty}\pi\left(m\right)=1$. Since $\pi\left(m\right)=\pi\left(0\right)\prod_{j=0}^{m}\frac{p_{j-1}}{q_{j}}$, that means $\pi\left(0\right)\sum_{m=0}^{\infty}\prod_{j=0}^{m}\frac{p_{j-1}}{q_{j}}=1$. This implies that $$0<\sum_{m=0}^{\infty}\prod_{j=0}^{m}\frac{p_{j-1}}{q_{j}}<\infty$$