Vtyjkyuk

Let

• $E$ be a countable set equipped with the discrete topology and $mathcal E:=mathcal B(E)$


• $pi$ be a probability measure on $(E,mathcal E)$ with $$pi(x):=pi(leftxright)>0;;;textfor all xin Etag1$$


• $q$ be a stochastic matrix$^1$ on $E$

Now, let $$p(x,y):=begincasesq(x,y)minleft(1,fracpi(y)q(y,x)pi(x)q(x,y)right)&text, if xne ytext and q(x,y)>0\0&text, if xne ytext and q(x,y)=0\1-sum_zin Esetminusleftxrightp(x,z)&text, if x=yendcases$$ for $x,yin E$.

It's easy to see that $p$ is reversible$^2$ with respect to $pi$ and hence $pi$ is stationary$^3$ with respect to $p$.

How can we conclude that, if $q$ is irreducible$^4$ and $$forall x,yin E:q(x,y)>0Leftrightarrow q(y,x)>0,tag6$$ then $p$ is irreducible?

I'm not sure if this is a simple fact following from $(3)$ or $(4)$, or if we we need to make use of the actual shape of $p$.

$^1$ i.e. $q:Etimes Eto[0,1]$ with $$sum_yin Eq(x,y)=1;;;textfor all xin E.tag2$$

$^2$ i.e. $$pi(x)p(x,y)=pi(y)p(y,x);;;textfor all x,yin E.tag3$$

$^3$ i.e. $$pi p(y):=sum_xin Epi(x)p(x,y)=pi(y);;;textfor all yin E.tag4$$

$^4$ i.e. $$forall x,yin E:exists ninmathbb N:q^n(x,y):=sum_zin Eq^n-1(x,z)p(z,y)>0,tag5$$ where $q^0(x,y):=1$ for all $x,yin E$.