A phi-irreducible chain can fail to converge from a pi-null start

The only nonempty $\varphi$-positive set contains state $1$. From $1$ it is already reached, and from $x\ge 2$ the one-step probability of reaching it is $x^{-2}>0$. Thus the chain is $\varphi$-irreducible. The point mass ${\delta }_1$ is invariant because state $1$ is absorbing. Since $\pi$ has only one positive-mass state, no partition into $d\ge 2$ classes all of positive $\pi$-mass exists; the chain is aperiodic under [F2].

Starting from $x\ge 2$, the probability of avoiding absorption through time $n$ is $s_n(x)={\prod }_{j=x}^{x+n-1}(1-\frac{1}{j^2})=\frac{x-1}{x}\frac{x+n}{x+n-1}\mathrm{.}$ The identity follows by writing $1-j^{-2}=((j-1)\mathrm{/}j)((j+1)\mathrm{/}j)$ and canceling adjacent factors. Thus $s_n(x)\to (x-1)\mathrm{/}x>0$, so the probability of ever hitting the $\varphi$-positive set $\{1\}$ is less than one. The chain is not Harris recurrent by [F3].

At time $n$, the mass outside $\{1\}$ is exactly $s_n(x)$. By [F5], $\mathrm{\parallel }{P}^n(x,\cdot )-{\delta }_1{\mathrm{\parallel }}_{\mathrm{T}\mathrm{V}}=\mathrm{\mid }{P}^n(x,\{1\})-1\mathrm{\mid }=s_n(x),$ which has positive limit. Therefore the stationary-a.e. convergence conclusion cannot be strengthened to all starts without Harris recurrence.