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.



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.

  • $\begingroup$ Did you mean to say can't? $\endgroup$ – Cehhΐro Oct 14 '16 at 6:58
  • $\begingroup$ Yes, thank you. $\endgroup$ – Math1000 Oct 14 '16 at 8:09
  • $\begingroup$ Thanks!! Just a follow up..the fact that any stationary distribution is multiple of invariant measure encompasses periodic markov chains I assume? It is just a result I am not familiar with in general.. $\endgroup$ – user1421195 Oct 14 '16 at 18:52

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