Limit distributions of sums of random variables defined on a finite homogeneous Markov chain. I (Q1897886)

Denote \(S_n(X, a) = \sum^n_{j=1} a(X_j)\), where \(a(i)\) denotes the \(i\)-th coordinate of the vector \(a \in R^n\), and \(X = (X_j)\), \(j \geq 0\), is a homogeneous Markov chain with a finite space having one class of communicate states. There are well-known conditions when \(S_n(X,a)/\sqrt {n}\) weakly converges to the normal random variable \(Z\) with \(EZ = 0\) and \(EZ^2 = \omega^2 \geq 0\). The author presents a detailed study of the case when the random variable \(Z\) is degenerated, i.e., when \(\omega = 0\). Some conditions are stated which are equivalent to \(\omega = 0\) and which, as the author supposes, are more convenient then existing ones. [For part II see below].