Limit \(\Gamma\)-distributions for sums of random variables, connected in an inhomogeneous Markov chain with two states
Let \(S_ n=x_{n1}+...+x_{nk_ n}\), where, for every \(n=1,2,...\), \((x_{ni})\) is a nonhomogeneous Markov chain with two states 0 and 1. The main theorem, given with proof, establishes necessary and sufficient conditions in order that the moment limit relations, implying the convergence of the distribution of \(S_ n/\sigma_ n\) to the a- parameter gamma distribution, hold. Here \(\sigma_ n=C\max_{i}T^+_ i\), \(T^+_ i\) and the conditions having explicit expressions in terms of the one step transition probabilities \(p^{(i)}\) of the n-th chain. The equivalence is proved under some restrictions: \(\sigma_ n\to \infty\), \(\max_{2\leq i}p_{01}^{(i)}\to 0\), \((\sum_{i}p_{01}^{(i+1)}T^+_{i+1})/\sigma_ n\to \delta\) (finite).
scientific article; zbMATH DE number 3909471