Limit distributions of finite Markov chains and semigroups of stochastic matrices (Q1366374)

We recall one theorem from the theory of summation of independent random variables [see \textit{W. Feller}, ``An introduction to probability theory and its applications''. Vol. 2 (1971; Zbl 0219.60003), Chapter IX, Theorem 2]: The following classes of probability distributions coincide: (i) limit distributions of sums \(S_{N,n(N)} =\sum^{n(N)}_{i=1} X_{N_i}\), where in each series random variables \((X_{N_i} \mid 1\leq i\leq n(N))\) are independent and identically distributed and the sequence of integers \(n(N)\) tends to \(+\infty\); (ii) distributions of increments of stochastic processes with independent and stationary increments; (iii) infinitely divisible distributions. We obtain the similar characterization of limit distributions of \(S_{N,n(N)}\) in the case where \((S_{N,i} \mid i\geq 0)\) is a finite Markov chain. The connecting link between the theory of summation of independent random variables and the theory of Markov chains is an operator approach in the theory of summation.