A Multi-Dimensional Limit Theorem for Homogeneous Markov Chains with a Countable Number of States

A homogeneous recurrent irreducible Markov chain with a countable set of states, $e_0,e_1,\dots,e_n$ is considered. The limit distributions for $n\to\infty$ of the random vector $N_n=(N_n^0,\dots,N_n^r)$ are investigated, where $N_n^r$ is the number of strikes during $n$ units of time in state $e_r$.
It is assumed that the distribution function $F(x)$ of the time for returning to the fixed state satisfies the condition that for any $c>0$,
$$\frac{1-F(cx)}{1-F(x)}\to c^{-\alpha},\quad x\to\infty,$$
for some $0\leq a<2,\alpha\ne1$. In this case it holds true that for some definite choice of affine transformations of an $(r+1)$-dimensional Euclidean space the distributions of vectors $A_n N_n$ converge to the undergenerate distribution on the $(r+1)$-dimensional space. The forming transformation $A_n$ and the characteristic functions of the limit distributions can be expressed.