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 $T$?