Let $X$ be a transient Markov chain with countable state space $S(X)$. Let $Y$ be a positive recurrent Markov chain with countable state space $S(Y)$. (Time is discrete.)


Let $A \subseteq S(X)$ be such that, with probability 1, $X$ visits $A$ infinitely many times. In fact, if it helps, let $A$ be such that, with probability 1, there exists a sequence $T\_n \to \infty$ such that $\frac{1}{T\_n}$(number of visits to $A$ during $[0,T\_n]$) $\to a > 0$. (Obviously such sets $A$ exist, e.g. $A = S(X)$.)


1) Is it true that, in fact, $\lim\_{T \to \infty} \frac{1}{T}$ (number of visits to $A$ during $[0,T]$) $= a$?


2) Let $y \in S(y)$ be any state; recall that $Y$ was positive recurrent. Is it true that the product chain $(X,Y)$ will visit $(A,y)$ with probability 1? Is there an estimate on $\frac{1}{T}$(number of visits to $(A,x)$ during $[0,T]$), at least for some sequence of $T$?



Product of a transient and a positive recurrent Markov chain
The answer to 1. is negative. Consider the transient Markov chain $X\_n=X\_0+n$ on $\mathbb Z$ and choose for $A$ any subset of $\mathbb Z\_+$ with different inferior and superior densities.


As regards 2., I assume you want $X$ and $Y$ to be independent. Then the answer to the first question is negative. Same $X$ as before, $X\_0$ deterministic, $Y\_n=(-1)^n$, $y=(-1)^{X\_0}$ and $A=2\mathbb Z+1$. Then $(X,Y)$ never visits $A\times\{y\}$.


