How do we prove that an irreducible, doubly stochastic (row and column sums to 1) discrete time markov chain with infinite state space cannot be positive recurrent?
Initial Ideas: I think the place to start is naturally trying to proving that the stationary distribution cannot possible exist for this setup...but I am having trouble getting started with proving this.
Thanks!
Let $P$ be the transition kernel and let $\nu= \mathbf 1$. Then for each $k$ we have $$(\nu P)_k = \sum_j \nu_jP_{jk} = 1, $$ so $\nu$ is an invariant measure for $P$. Since $P$ is irreducible, any stationary distribution for $P$ must be a multiple of $\nu$. But $\sum_k \nu_k=\infty$, so there cannot be such a distribution.