Limit distributions of sums of random variables defined on a finite homogeneous Markov chain. II (Q1897902)

[For part I see above.] Here the author considers the random variable \(S_{n,N} = \sum^n_{j = 1} a_N (X_{N_j})\), where now \(a_N(i)\), \(N \geq 1\), denotes the \(i\)-th coordinate of the sequence of vectors \(a_N \in R^n\), and \(X_N = (X_{N_j}\), \(j \geq 0)\), \(N \geq 1\), is the series of homogeneous Markov chains with a finite space \([d] = \{1, 2, \dots, d\}\) and irreducible transition matrix. All possible limit distributions of the appropriately centralized and normalized sums \(S_{n,N}\) are described.